J he Person char • * t 
 ,"""• ~™-J m, ' c ' 1 ' '"•'«»• befo " (he 
 
 re nevy co// - "i dismijj., , 
 
 LJ61- 
 
 O-J096 
 
Digitized by the Internet Archive 
 
 in 2012 with funding from 
 
 University of Illinois Urbana-Champaign 
 
 http://archive.org/details/stochasticanalysOObull 
 
ENGINEERING LIBRA 
 UNIVERSITY OF ILLINC 
 
 URBANA. ILLINOIS 
 
 Center for Advanced Comnu 
 
 UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN 
 
 URBANA. ILLINOIS 61801 
 
 CAC DOCUMENT NO, 208 
 
 STOCHASTIC ANALYSIS OF UNCERTAINTY 
 IN A U.S. INPUT-OUTPUT MODEL 
 
 . by 
 
 Clark W. Bullard 
 Donna L. Amado 
 Dan L. Putnam 
 Anthony V. Sebald 
 
 September 1976 
 
CAC DOCUMENT NO. 208 
 
 STOCHASTIC ANALYSIS OF UNCERTAINTY 
 IN A U.S. INPUT-OUTPUT MODEL 
 
 by 
 
 Clark W. Billiard 
 Donna L. Amado 
 Dan L. Putnam 
 Anthony V. Sebald 
 
 Center for Advanced Computation 
 
 University of Illinois at Urbana-Champaign 
 Urbana, Illinois 
 61801 
 
 September 1976 
 
Table of Contents 
 
 Page 
 
 1.0 INTRODUCTION .'.... '. ...... 1 
 
 2.0 DATA BASE PREPARATION ' '3 
 
 2.1 The Model . ......'. ' . . 3 
 
 2.2 The Data i| 
 
 3.0 METHODOLOGY. ......................... 6 
 
 3.1 Point of View for Stochastic Error Analysis . . ...... . -6 
 
 3.2 Sampling Random Variables . . 7 
 
 3.3 Aggregating Random Variables. . . : 11 
 
 3.^ .Results Saved for Analysis 15 
 
 3.5 Stopping R\il e . . . . . ... .... . . '. .• . .17 
 
 3.5.1 An Outline of the Approach . , . 18 
 
 3.5.2 A Mathematical Overview. ..... 19 
 
 3.5.3 An Example . . . ... . . . . . . . . . . . . . . . 25 ' 
 
 3.5.^ Mathematical Derivations .............. 27 
 
 k.O ANALYSIS OF RESULTS 29 
 
 k.l Goodness of Fit . . . 29 
 
 ^4.2 Confidence Intervals 31 
 
 k.3 Variability of the Elements in the Result Set 32 
 
 k.h Bias on Elements of the Result Set. 32 
 
 U.5 Sensitivity of Simulation Results to Assumptions on 
 
 Input Uncertainties 3^ 
 
 h.6 The Effect of Aggregation 36 
 
 4.7 Results for the 101 Sector Model. 37 
 
 ERRATA NOTTCF: Pages 32 - 36 were inadvertently incorrectly numbered, 
 The continuity of the text is correct, 
 
Table of Contents (continued) 
 
 Appendix A: BASE YEAR UNCERTAINTY ESTIMATES. .' • . . 39 
 
 A.l BEA DATA , ' 39 
 
 A. 1.1 INTRODUCTION 39 
 
 A. 1.1.1 Sources of Error ' . . 1*0 
 
 A. 1.1. 2 Effects of Scale '...•... 1*1 
 
 A. 1.2 METHOD . . . . .1*2 
 
 A. 1.2.1 Quantities Estimated. ............ "^2 
 
 A. 1.2. 2 Degree of Detail • . . 1*2 
 
 A. 1.2. 3 Interview Techniques ; . 1*3 
 
 A.1.2..1* Bias. . . . " kk 
 
 A. 1.2. 5 Effect of Numerical Magnitude hk 
 
 A. 1.3 UNCERTAINTY ESTIMATES 1*5 
 
 A. 1.3.1 Direct Allocations 1*5 
 
 A. 1.3. 2 Gross Domestic Output 1*9 
 
 A. 1.3. 3 Transfers 1*9 
 
 A.l. 3.1+ Margins 50 
 
 A.l.l* CONCLUSIONS, APPLICATIONS, AND LIMITATIONS 50 
 
 A. 2 DIRECT ENERGY ALLOCATIONS 52 
 
 A. 3 DISPERSION FACTORS FOR SMALL-MAGNITUDE FIGURES 58 
 
 A.l* PROBABLE VALUES OF "ZERO" ELEMENTS 59 
 
 Appendix B : SECTOR DEFINITIONS 60 
 
 Appendix C: TREATMENT OF ZERO VALUES 62 
 
 C.l VARIANCE OF A "FOLDED" NORMAL DISTRIBUTION 62 
 
 C.2 VARIANCE OF A LOGNORMAL DISTRIBUTION 62 
 
Table of Contents (continued) 
 
 Appendix D: RANDOM NUMBER GENERATOR VERIFICATION 
 
 D.l INTRODUCTION 6k 
 
 D.2 INDEPENDENCE BETWEEN MONTE CARLO SAMPLES ...... 67 
 
 D.3 INDEPENDENCE BETWEEN INPUTS TO A MONTE CARLO SAMPLE' 73 
 
 D.U TESTING FOR NORMALITY . .77 
 
 REFERENCES 83 
 
1.0 INTRODUCTION 
 
 This report describes a stochastic parametric sensitivity analysis of 
 a detailed structural model of the U.S. economic system. Model parameters 
 defining the structure of the system at the time of measurement were derived 
 from physical observations of the system. Use of such models is becoming 
 increasingly prevalent for mid-to long-range studies and policy analyses in 
 government planning at all levels. Resource scarcity, foreign policy con- 
 tingencies and other factors have made rapid structural change the object of 
 analysis, not something one can assume avay. Effective use of such models 
 requires an understanding of the effects of parametric change and uncertain- 
 ty. 
 
 We are concerned here with a linear static input-output model of the 
 U.S. economy. Its parameters are derived from data on interindustry trans- 
 actions complied by the U.S. Department of Commerce. Due to the 
 size and complexity of the economic system, funding limitations and measure- 
 ment lags, these parameters are seven years out of date when published. 
 Parametric uncertainty therefore can arise from two sources: observation 
 of the system during the base year and structural changes during the seven 
 year lag period. Estimates of uncertainty in the base year parameters were 
 compiled by Bullard (1976) and are discussed in Appendix A. 
 
 The effect of parametric uncertainty on model outputs has been dis- 
 cussed by Sebald (197M and Bullard and Sebald (1975). These papers 
 quantified the maximum error tolerances that would result from the worst- 
 case distribution of parametric errors. For this model, it was found that 
 the process of matrix inversion could magnify input errors by more than 
 
emphasizing the need for developing a methodology that could 
 quantify the extent to which parametric errors cancel one another. 
 
 The Monte Carlo simulation analysis described here was designed to 
 answer that question. Base year interindustry transactions were character- 
 ized as random variables ana the model parameters were derived from them. 
 The results from each simulation were used to update a set of sufficient 
 statistics to yield unbiased estimates of means, variances and some covari- 
 ances. The simulations were performed to evaluate both the effect of doubling 
 error tolerances on inputs and the effect of changing the structure of the 
 model to enhance its usefulness for predictive work. 
 
 After 200 simulations, the preliminary results were analyzed in order 
 to determine the cost-effectiveness of proceeding with more simulations. 
 In all cases, this a priori determination of the confidence intervals on 
 final results showed that 1000 runs would be adequate. These estimates were 
 then verified when the simulation had been completed. 
 
 Chapter 2 describes the preparation of the data base and estimation of 
 uncertainty on base year transactions. Chapter 3 details the simulation 
 methodology, the criteria for determining "acceptability" of simulated 
 parameters and derivation of the stopping rule. Chapter h presents the 
 results of all simulations, and discusses the effects of aggregation, 
 magnitude of input uncertainty and other <. .riables. 
 
2.0 DATA BASE PREPARATION 
 
 2.1 The Model 
 
 The linear static input-output model of the U. S. economic system is 
 described in detail by Bullard and Herendeen (1975). It is based on the 
 theory developed by Leontief (19^1), and relies largely on data assembled by 
 the U. S. Department of Commerce, Bureau of Economic Analysis (BEA). Data 
 are expressed in constant dollars, 'which act as a surrogate for physical 
 units. In this particular model however, the inputs of energy to all sectors 
 are expressed in physical units, to take account of the fact that energy is 
 sold to different users at different prices. 
 
 The governing equation of the model is 
 
 (I-A) X = Y (2.1-1) 
 
 where X is an N-order vector of gross domestic outputs for each sector, Y is 
 the vector of total final demands for the output of each sector, and A is the 
 matrix of parameters describing the technology of producing goods and services 
 during the base year. A typical element A. . represents the amount of input 
 from sector i required directly by sector j to produce one unit of its out- 
 put. These parameters are derived from base year observations of interindustry 
 
 transactions, T.., (amount of output from sector i sold directly to sector j): 
 ■*- J 
 
 T. . 
 A E -iJ- (2.1-2) 
 
 In turn, these interindustry transactions are defined as the sum 
 
 T = DA + MDT + TF (2.1-3) 
 
where DA is the amount of product i sold directly to sector J, MDT. 
 1 J ij 
 
 represents the transportation or trade margin i on all inputs to sector j , 
 
 and TF. . represents the amount of product j produced as a secondary output 
 -'-J 
 
 by sector i. 
 
 2.2 The Data 
 
 Estimates of all elements of the above matrices are collected and 
 assembled by BEA at the kQk sector level of detail. Before publication 
 however, they are aggregated to about 360 sectors. BEA personnel respon- 
 sible for this compilation were interviewed; their subjective estimates of 
 uncertainty on all base year transactions are given in Appendix A. 
 
 Before proceeding with the Monte Carlo simulation, these data were 
 aggregated to 90 and then to 30 sectors. The 30 sector data base was used 
 for the development and verification of all the computer programs. The 90 
 sector data base was used for the main simulation. This degree of aggregation 
 was chosen for economic reasons (matrix inversion is an expensive N 
 operation) and because it corresponds most closely to the most widely dis- 
 tributed and used version of the BEA input-output tables. These are published 
 at the 83 sector level of detail, while the 90 sector model used here retains 
 more detail in the transportation and energy sectors of the economy. 
 
 Aggregating an input-output data base is a nontrivial operation since 
 it must be done prior to the operations in (2.1-3). After aggregating the 
 
 Names of sectors at each level of aggregation are given in Appendix B. 
 

 three matrices independently and summing to obtain T, X and A are computed 
 using eqs. (2.1-1) and (2.1-2). 
 
 A 101 sector data base was also constructed by replacing the 5 energy 
 sectors in the 90 order model by 16 energy . supply and service sectors. 
 The rationale and development of the 101 sector model are described by 
 Bullard and Sebald (1975). Its purpose is to more accurately predict energy 
 consumption in future years by explicitly modeling fuel substitutability. 
 In effect, this model recognizes that end uses of energy (space heating, 
 lighting, air conditioning, etc.) are less substitutable than the fuels 
 themselves by permitting the non-energy sectors to purchase only end uses 
 of energy, while fuels are sold only to the end use sectors. Note that 
 the former are very stable over time while the latter are not. The most 
 variable coefficients or parameters involved in energy consumption are 
 thereby confined to the few representing sales of fuels to the end use 
 sectors, which can be estimated independently using models designed 
 expressly for that purpose. 
 
 * Names of sectors at each level of aggregation are given in Appendix B, 
 
3.0 METHODOLOGY 
 
 3.1 Point of Viev for Stochastic Error Analysis 
 
 There are several ways to interpret this problem, and the point of view 
 affects both the methodology and the interpretation of results. One way is 
 to act as a simulator of BEA's activities from data collection through matrix 
 inversion. In an alternative viewpoint, the analyst attempts an a priori de- 
 termination of the effect of mathematical transformations on uncertain obser- 
 vations. In either case, this information enables the analyst to assess the 
 usefulness of the data for modeling purposes. We have adopted the latter point 
 of view. 
 
 Within this framework, the analyst receives signals from the economic 
 system associated with each interindustry transaction as well as total output 
 and value added. Actually, each of these signals from an industry is the sum 
 of many signals from individual establishments. The signals appear to be in- 
 dependent; that is to say, the signals tell us little about their correlation.* 
 The analyst's only information on these correlations comes from accounting 
 identities requiring income to equal outgo. 
 
 Each signal is characterized by BEA in terms of upper and lower bounds and 
 a "published value" representing their estimates of where the true value is mos 
 likely to lie. We then characterize BEA's knowledge of the transactions as ran 
 dom variables. The distributions are inputs for the Monte Carlo analysis which 
 transforms them into a set of numbers comprising the solution set. Each elemen 
 of the solution set is characterized by a set of statistics** which are then 
 compared with the deterministic result. 
 
 * Due to the size and complexity of the economic system, frequent measurement 
 
 is economically prohibitive so no information is available from time series 
 analysis. 
 
 ** Mean, variance 6 
 
Each input variable is first sampled independently, but some effort is 
 then made to assure that the external balance conditions are satisfied. This 
 is what BEA does in their deterministic approach, and we make a similar 
 attempt with our Monte Carlo approach. It is unrealistic however, to 
 completely simulate BEA's activities, many of which are judgemental, undocu- 
 mented, and not reproducible. The specific shortcuts taken are detailed in 
 section 3.^. 
 
 3.2 Sampling Random Variables 
 
 All of the basic data (transactions, industry output, final demands) 
 are characterized as random variables having either normal or lognormal 
 distributions.* As discussed in Appendix A, Section k t entries which 
 
 have been truncated to zero by BEA are modeled with a "folded normal" ran- 
 dom variable, which is simply the absolute value of a normal random variable 
 with mean 0. Non-zero cells are modeled using either normal . or lognormal 
 random variables with the former used in those cases where the published 
 value is relatively accurate. In situations where the data is less well 
 known, an analyst will tend to use a multiplicative factor to bound his 
 estimate rather than an additive error bound. A lognormal distribution is 
 appropriate in such a case because of its property of multiplicative sym- 
 metry about the median. That is, if X_is the median of a lognormal random 
 variable X, then Prob. (X > X D) = Prob^ (X < X /D) for any factor D. For 
 
 * In a few cases a negative entry in the data is modeled by the negative of 
 a lognormal random variable (which necessarily takes only positive values). 
 This set of circumstances is handled so much like the usual lognormal case 
 that it is not discussed separately in what follows. 
 
example, if an analyst states that his estimate has probability a of being 
 correct within a factor of D, then a lognormal random variable with a = 
 Prob. (X fl /D < X <_ X D) will be used to model the situation. 
 
 This section outlines a procedure for sampling from random variables 
 such that 
 
 1) The sample will be drawn from a folded normal, normal or lognormal 
 population. 
 
 2) The distributions will be truncated to prevent samples that are 
 absurd (e.g., negative transactions). Truncation eliminates 
 samples in the upper and lower 0.15$ tails in the normal and log- 
 normal cases and in the upper 0.3$ tail in the folded normal case. 
 This corresponds to the percentage of probability outside 3 
 standard deviations from the mean in a normal population. 
 
 3) The expected value of the sampled result is equal to the published 
 value, M, of the entry in question (except in the folded normal 
 case where the published value is zero). 
 
 k) Before truncation, the random variable X from which we sample 
 has a confidence interval defined by a parameter b, 5 or D. 
 
 a. Folded Normal Case 
 
 Prob (X < b) = .997 
 
 (i.e., b amounts to 3 standard deviations of the underlying 
 
 normal random variable. 
 
 b. Normal Case 
 
 Prob (p - 6 ^x- X - y JC + 6 V = '" 7 
 
 (i.e., o amounts to 2> standard deviations of X expressed as a 
 
 fraction of the mean, M y = M) 
 
 c. Lognormal Case 
 
 Prob (X Q /D < X < X Q D) = .997 
 
 In all three cases the sampling procedure is based on a standard normal 
 random variable (i.e. , mean = and variance = 1, denoted N(0,l)). 
 
 * The standard normal random number generator used was the International 
 Mathematical Statistical Library routine GGNRF. Tests of randomness and 
 normality were performed for verification purposes and are described in 
 Appendix D. 
 
Truncation is achieved by sampling until a value r is obtained which is 
 less than 3 in absolute value. In the folded normal case we set y = and 
 a = b/3 so that y + or is a sample from a truncated N(y, a ) variable; the 
 
 absolute value then satisfies the conditions for the folded normal sample. 
 
 6M 
 In the normal case we set y = M, a = — and then y + ra is used as the 
 
 normal sample. 
 
 The situation for X lognormal is slightly more complicated. In this 
 
 Y 
 case, X = e where Y is a normal random variable. Let y and a be the mean 
 
 and standard deviation of Y. 
 
 Then the median X of X is equal to e so y = In X . Therefore, 
 
 Prob (X Q /D < X <_ X Q D) = .997 
 implies that 
 
 Prob (lnX - InD <_ Y <_ lnX Q + InD) = 
 
 Prob (-InD <_Y - y <_ InD) = .997 
 so that 
 
 InD = 3a. 
 
 Since we want the mean value to equal the published value, 
 
 2 
 
 y + 7T . „ a n ,, In D 
 
 y = e 2 = M, we must set y = InM - — = InM - n 
 
 To summarize, we sample for X in the lognormal case by obtaining a 
 
 truncated standard normal random number, r, and setting X = e where 
 
 2 
 a = InD and y = InM - n . Comparing the three cases we have; 
 
 X Folded Normal X Normal X Lognormal 
 
 y = 
 a = b/3 
 
 y = M 
 a = 6M/3 
 
 y = InM - 
 a = lnD/3 
 
 2 
 In D 
 
 18 
 
 X = truncated ABS(N(y,a )) X = truncated N(y,o ) 
 
 X = truncated e 
 
 N(y,a' 
 
In the lognormal case the mean is not coincident with the median. To 
 evaluate the error resulting from assuming they are equal, suppose an analyst 
 gives a confidence interval for the true value T, in terms of his estimate 
 M and a factor D. That is, 
 
 Prob (M/D < T <_ MD) = .997 
 where .997 is just the probability spanned by three standard deviations 
 about the mean in a normal distribution. 
 
 We have modeled this situation with a random variable X with 
 u = M and 
 
 Prob (X /D <_ X < DX Q ) = .997 
 We want to show that 
 
 Prob (M/D < X 1 DM) is close to .997- 
 In fact, 
 
 Prob (M/D < X < DM) = 
 
 Prob (inM - InD < Y < InD + InM) = 
 
 2 2 
 
 Prob (y+|- -3a<Y<3c+y+|-)= 
 
 Prob (§- 3 <_^< 3 + £) 
 d. a 2 
 
 Y-u 
 
 Since is standard normal, we can find this probability in standard 
 
 normal tables if we know a. For a typical value of D such as 
 2 
 
 o o _ InD 
 D = 8, - = - T ~- = .35 
 
 Therefore, 
 
 Prob (M/D <_ X < DM) = Prob ( . 35 - 3 < — < 3 + .35) = -995 , 
 
 a _ ' 
 
 indicating that the error resulting from the assumption is negligible 
 
 .10 
 
3.3 Aggregating Random Variables 
 
 Based on subjective uncertainty estimates made by BEA Personnel, 
 probability distributions were defined at the 360 sector level of detail. 
 Since simulations were done at the 101, 90 and 30 sector levels, aggregation 
 was necessary. The means of the aggregated variables are easily obtained 
 but specification of the distributions of the aggregate variables is a non- 
 trivial task which was undertaken in the following way. Since all trans- 
 actions, margins, etc. at the 368 order are in fact aggregates of data 
 obtained initially from individual establishments grouped by 5 or 6 digit 
 Standard Industrial Classification codes, the specification of a distribution 
 for these aggregates was a crude assumption in itself. The basis for 
 specifying the distribution at the 90 sector level is equally subjective, 
 so we adopt the following convention. Assume that the variance, V, of 
 each aggregated element is the sum of the variances of all its constituents. If 
 3/"v is less than hQ% of the aggregated mean, y, assign a normal N('y,V) 
 distribution to the variable. If 3"V is greater than k0% of y , a lognormal 
 distribution is assumed. If y equals 2iero, a folded normal distribution is 
 used. This rule is simply a formalized reproducible characterization of a 
 subjective assessment of input data uncertainty. It is felt that the sub- 
 jective nature of the disaggregated uncertainty estimates did not warrant a 
 more rigorous approach. For purposes of reproducibility, however, the 
 adopted algorithm is detailed below. 
 
 The first step in aggregating is to compute the variances of the entries 
 being aggregated: 
 
 11 
 
* 
 
 Case 1: Folded Normal. V = (b/3) 2 (l- -) 
 
 TT 
 
 Case 2: Normal. V = (M 6/3) 2 
 
 * 2 1 2 D 
 
 Case 3: Lognormal. V = M exp( — — ) - 1 ! 
 
 As indicated above, the variances of the entries being aggregated are 
 summed to obtain the variance of the aggregate entry. The decision to make 
 the aggregate entry folded normal, normal or lognormal depends on the ag- 
 gregate variance V and the aggregate published value M. 
 
 Case 1: M = 0. Here we assume that the constituent entries were 
 
 also published zeros. The parameter b is chosen so that the 
 variance of the resulting folded normal will e qual t he give n 
 aggregate variance V. This value is b = / V/(l - 2/tt) 
 
 Case 2: M 4 and 3 / V/ IM1 <_ .k. These entries are modeled as normal 
 with 6 chosen so that the resulting variance will equal V. 
 This value for 6 is 6 = 3/"v?!M| 
 
 Case 3: M ^ and 3 / V/ [Ml > . U. Here we use a lognormal random 
 
 variable specified b y the paramete r D chosen to be consistent 
 withv. D = exp(3 / ln(l + V/M^))* 
 
 3.3 Constructing the Transactions Matrix 
 
 Fig. 3-1 shows graphically the relationship between the matrices of 
 
 transactions, (T), final demand (FD), imports (M) and gross domestic outputs 
 
 (GDO). 
 
 N 10 
 
 T T. + J FD. n - M. = GDO. (3.3-1) 
 
 j=l 1J k=l lk X 
 
 These random variables are sampled from normal or lognormal distributions as 
 described above. Each element in the first row (i=l) is sampled first 
 independently, just as BEA analysts receive these values from apparently in- 
 dependent sources. Since eq. (3.3-1) is an external balance condition that 
 
 *See appendix C for details. 
 
 12 
 
o 
 o 
 
 X 
 
 X 
 
 o 
 
 -P 
 
 o 
 
 a 
 o 
 
 •H 
 
 -p 
 
 O 
 
 w 
 
 u 
 
 En 
 
 !h 
 
 cd 
 
 
 >H 
 
 <L> 
 
 pq 
 
 H 
 
 i 
 on 
 
 •H 
 
 13 
 
is not satisfied in general, we force this condition to be satisfied in 
 
 much the same manner as BEA does. The lognormally distributed variables in 
 the row are generally those obtained from unreliable sources or computed 
 
 using surrogate variables. Therefore these values are scaled proportionately 
 
 * 
 
 to satisfy eq. (3.3-1). 
 
 Proceeding in this manner through N rows, a complete data set is con- 
 structed satisfying row constraints. The rows are not independent, however, 
 because the value of all outputs (GDO) of a sector must equal the value of 
 all commodity inputs (from the other N sectors) plus "value added" (a term, 
 VA, accounting for wages, taxes, and profit). VA is measured independently 
 by federal agencies and provides BEA analysts with another external 
 
 condition to satisfy. Their method for satisfying this was too complex to 
 
 »* 
 
 model, so a simpler check had to be devised for this Monte Carlo study. 
 
 The method employed is based on the response of the BEA's director 
 
 of the 1-0 study to the following question: "If the criterion for terminating 
 the iterative process of balancing the 1-0 table were based on uncertainty of 
 the VA values, how much could be tolerated?" The answer indicated that out 
 of 90 sectors, at least 88 must be within ±20$ of the "true" value.*** If 
 the condition was not met, the matrix was rejected. This condition was never 
 violated in the actual simulation. 
 
 * In fact, BEA analysts actually estimate many of these uncertain values by 
 computing the difference between GDO and the sum of the well known (normall; 
 distributed) variables and allocating proportional to some surrogate 
 variables (e.g. employment). 
 
 ** In the 1967 input-output study, consistency between row and column sums 
 
 was assured by assigning responsibility for individual sectors to different 
 analysts and after each independently estimated initial row values, the 
 resulting columns were presented to each analyst for independent verifi- 
 cation. After many iterations and some undocumented Judgement decisions, 
 the "published" values were agreed upon. 
 
 *** Philip M. Ritz (1976) Interindustry Economics Branch, Bureau of Economic 
 Analysis, U. S. Department of Commerce, personal communication. 
 
 Ik 
 
Next, the terms in eq. (3.3-1) are used to compute the coefficients 
 
 T. . 
 
 A = -±d- 
 ij GDOj 
 
 and the Leontief inverse matrix (i-A) " is finally calculated. Aside from 
 checking the eigenvalues of A, there is no a priori check that can he performed 
 to guarantee positivity of the inverse matrix. Therefore, each inverse matrix 
 is checked after it is computed to verify that every element is greater than 
 zero. If it fails the test, all the randomly selected variables T, FD, M, 
 GDO are discarded and a new set is selected. This is exactly the procedure 
 employed by BEA. Again, the simulation was completed without this condition 
 being violated. 
 3.^ Results Saved for Analysis 
 
 The simulation described here is expensive from a computational point 
 
 3 
 of view since matrix inversion is an N operation. For this reason, every 
 
 simulated Leontief inverse matrix was saved on tape so it would be available 
 
 ** 
 
 for future analysis if necessary. 
 
 For purposes of this analysis, our attention was focused on the means, 
 
 variances and confidence intervals for the elements of (i-A) and selected 
 subsets and linear combinations thereof. To calculate these, it was necessary 
 to save a set of sufficient statistics on disk after each iteration, the 
 
 running sum and the sum of the squares for each element of the following set 
 of results which we shall denote by fi: 
 
 * If all variables were expressed in current-year dollars, some a priori 
 tests are available. In the general case such as thin one, where the 
 energy sector outputs are uxprt-ujjtnJ in [Aiy \: i caJ unit.;:, no awAi tests exist 
 
 * * The tape will be delivered to EPRI under separate cover. 
 
 15 
 
1. The entire (i-A) matrix; 
 
 2. The total primary energy intensity vector, e; and 
 
 3. The sector output vector, X. 
 
 The total primary energy intensity vector is a linear combination of the 
 
 energy rows of (i-A) , and a typical element e. represents the amount of basic 
 
 J 
 
 energy resources required directly and indirectly to produce one unit of output 
 from sector j for final consumption. The sector outputs X are computed from 
 the simulated (i-A) ' matrix using the base year domestic final demands as 
 weighting factors: 
 
 1 10 
 X. = I (I-A) 7. ( I FD. . - M.) (3.U-1) 
 
 This is done because 1-0 models are frequently employed to estimate total sectc 
 outputs corresponding to a specified final bill of goods, and a significant 
 amount of additional error cancellation may be achieved. 
 
 In order to ascertain the nature of the distribution of typical random 
 variables, each simulated value was saved for source results. The 
 variables saved were X, e, and the electricity sector row of (i-A) . Goodness 
 of fit tests performed on these variables are described in Section h. 
 
 Finally, since most applications of the particular models examined are in 
 the area of energy policy analysis, it was decided to save sufficient stat- 
 
 * The energy rows utilized are those corresponding to coal, crude oil and gas, 
 and the fossil fuel equivalent of hydro and nuclear electricity: e. = (i-A). 
 
 + (I-A)" 1 + 0.6 (I-A)" 1 !*. 
 
 16 
 
istics for recovering covariances of the energy sector rows of (l-A)~ . Since 
 all possible linear combinations were ~ot o^ interest - only row and column 
 combinations - storage requirements were considerably reduced. It was sufficient 
 to save the running sums of products of all pairs of entries appearing together 
 in such linear combinations. If other combinations are ever needed, they will 
 be recoverable from (i-A) matrices saved on an archive tape as described 
 earlier. 
 
 With this set of results it is possible to estimate the total energy require- 
 ments to meet arbitrarily specified final demands, and to compute linear 
 combinations of energy intensities similar to the "total primary" one described 
 earlier. 
 
 3.5 Stopping Rule 
 
 One of the major difficulties associated with Monte Carlo simulation is 
 knowing how many runs will be required to attain reasonable confidence in- 
 tervals on the results of the simulation. There are two major problem areas. 
 If one is considering whether or not to use Monte Carlo techniques, an 
 estimate of the required number of runs is crucial to determination of 
 simulation costs. It may be, for example, that reasonable confidence in- 
 tervals may require a prohibitively expensive number of runs. The second 
 problem arises after the decision has been made to use Monte Carlo methods. 
 One needs to know when enough runs have been made. 
 
 In the first problem area, present practice dictates running several 
 small scale simulations of a similar nature to the one of interest in order 
 to be able to extrapolate the number of runs in the smaller cases to the 
 probable runs needed in the larger. In the second area, good statistical 
 
 IT 
 
practice dictates that before taking any samples, one must determine how to 
 stop sampling in a way that doesn't bias results. Executing additional runs 
 if the resulting confidence intervals are too large is considered unwise 
 since one runs the risk of biasing the simulation results by stopping when 
 the desired outcome occurs. 
 
 In this section we present a method for determining, based on a very 
 small number of runs, the proper number of total runs the simulation should 
 require. The method properly elucidates the cost/benefit tradeoff between 
 the cost of additional runs and the benefits of increased accuracy. Informa- 
 tion is displayed to the analyst in a way that facilitates his making a 
 judgement on the proper number of runs to be made. Since this method 
 is based on just the first few runs, biasing of the simulation is not a 
 problem. Based on a very small number of runs, it is also a cost effective 
 way to decide whether a Monte Carlo analysis is economically feasible. 
 
 In section 3.5.1 we outline the approach used, in section 3.5.2 a 
 brief sketch of the mathematics involved is given, followed by an example 
 in section 3.5-3. Mathematical derivations are given in section 3.5.^. 
 
 3.5.1 An Outline of the Approach 
 
 Suppose a relatively small number of simulation runs have been made and 
 unbiased estimates of the second order statistics of all elements of a set 
 of results, ft, have been calculated. Since the estimates are themselves 
 random quantities, one can determine an interval about each estimate which 
 contains the population value (e.g. mean or variance) with a certain probabilit 
 These intervals are called confidence intervals (Cl) and we shall interest 
 
 18 
 
ourselves in intervals for which the corresponding probability is .95. 
 Fig. 3.5-1 describes the situation within which one must interpret the 
 results of a simulation. Generally, then, the simulation output is given in 
 two ways: the unbiased estimates are tabulated and their confidence intervals 
 (e.g. 95%) are also given. Our strategy will be to make a few runs and then 
 based on the resulting estimates and CI's, determine how many runs the entire 
 simulation will require. Two major effects occur with increasing sample size. 
 First, the estimates y and a will move around, ultimately converging to the 
 correct values. Second, the width of the CI's will decrease monotonically to 
 zero as the number of runs goes to infinity. For purposes of the stopping 
 rule, we have chosen to quantify the resulting simulation accuracy by monitor- 
 ing a histogram of 
 
 T3+4 3o + U 
 
 ■d - — 
 
 y 
 
 for each stopping point considered. The algorithm therefore: 
 
 1) Draws a histogram of the actual B values after an initial number of 
 runs , m . 
 
 2) On the basi| of the information after m runs, draws a histogram of 
 predicted B values after ^ 2 runs, where m 2 > m denotes a possible 
 stopping point for the full blown simulation. 
 
 3) Step 2 for various m . 
 
 Typical results are displayed in fig. 3.5-2. 
 
 3.5.2 A Mathematical Overview 
 
 As before, we denote the matrix of simulation output variables by ft. Each 
 X e ft has an unknown distribution which is at best only approximately normal. 
 
 The symbols are defined in Figure 3. 5 - l. 
 
 19 
 
A 
 IT 
 
 ■A. 
 
 x 
 
 a. b 
 
 f44 
 
 1 
 
 Figure 3.5-1 Mean and Variance Estimates and their Confidence Intervals 
 
 u « unbiased estimate of the mean of the underlying distribution a and b are 
 the upper and lower confidence interval lengths for y. 
 
 - unbiased estimate of the standard deviation of the underlying distribution 
 
 U and L are the upper and lower confidence interval lengths for 3o. 
 
 20 
 
ho 4o s ^ 
 
 ™ SO 
 
 °/ cr{ -tli^wu- *4 * 
 
 Figure 3.5-2 Stopping Rule Output Histograms for the 90 Order 
 Simulation. 
 
 21 
 
We denote the set of samples of each X e ft by {x.}. Although what follows 
 could be done for the unknown density of X e ft , this approach involves inac- 
 curacies in the determination of the fourth central moment and is computationa] 
 quite expensive. Instead we have chosen to convert each X e ft to a normal 
 random variable Z by the transformation 
 
 lOi 
 
 1 10 j = 10(i-l) J 
 
 Since the samples x are independent and identically distributed (iid) the 
 
 J 
 
 2 
 
 central limit theorem implies that the z. are approximately N(y ,o /10 ) . All 
 
 i y •* 
 
 statistical evaluations will be performed on Z and the results will be back- 
 transformed via (3.5-1) to X. Since the Z's are normal, the unbiased estimates 
 for their mean and variance are given by the well known relations [Winkler 
 & Hayes (1970)] 
 
 V = n 
 
 i=l 
 
 y = - I z. (3.5-2) 
 
 n .^ 1 
 
 -.2 
 
 I (z.- M V 
 
 o = i=± (3.5-3) 
 
 (n-1) 
 
 where n = m/10 is the number of Z sample points and m is the total number of ru 
 We assume that even after the initial 1^ runs, the CI around J is very small. 
 
 This has empirically been verified as a valid assumption and it permits us to 
 evaluate B+ for the Z variables by only worrying about the upper CI on a which 
 
 "2. ~ o 
 
 we denote by a . Again due to the normality of the Z's, c is well known to 
 u u 
 
 be [Winkler & Hayes (1970)] 
 
 ; 2 (n-l) a 2 
 
 (.975; n-l) 
 
 22 
 
■where ■)(• / _n denotes the value in a chi-square distribution with n-1 
 J 
 
 degrees of freedom cutting off the upper .975 of sample values. 
 B for the Z variables is then given by 
 
 3o T 
 
 B + = _Un = ^1 / (n-1) (3>5 _ 5) 
 
 n y y / v 2 
 
 n V * (.975. n-1) 
 i 
 
 where the subscript n denotes the number of Z sample points in the simulation. To 
 
 evaluate B we shall require (J and a . Even with relatively small n, 
 n 2 n 2 n 2 
 
 y ^ p since the CI even at n, is very short. Such is not the case with 
 n.,?* n 1 
 
 ** 
 
 a . We can, however, upper bound a if a is known by noting that 
 n 2 n 2 n ± 
 
 2~ 1~ 1 has an F distribution with n - n and n degrees of freedom 
 (n 2 -n 1 )(n 1 -l) 
 
 o 
 
 n 2 
 where Q ^ ttt~. We can then determine K such that 
 
 a 
 
 n l 
 
 P{Q <_K} = .50 (3.5-6) 
 
 and then use 
 
 a = A o (3.5-7) 
 
 n 2 n x 
 
 in (3.5-5). It is clear from (3. 5-5) and (3.5-7) that for r^ and n 2 fixed, 
 the B + is linearly related to B for each Z. The histogram of Figure 
 
 n 2 n ± 
 
 + 
 
 3.5-2 can then be generated by evaluating the actual histogram for B after 
 
 * 
 
 N. = M./10 
 
 i : 
 ** 
 
 This is proved in section 3.5-^. 
 
 23 
 
n runs, fixing n , evaluating the constant multiplier, C, and multiplying 
 each point in the histogram by C. In particular, 
 
 B = C B + Where 
 
 n 2 n ± 
 
 C = / K 
 
 n 2 " X \ / n i - 2 
 
 $ (.975-, n 2 -l)y^# (.975; r^-l) 
 
 and K is given by (3.5-6) 
 
 The choice of the probability .5 in (3.5-6) is not arbitrary. We a^e 
 
 interested in predicting the histogram of B from the histogram B Foi 
 
 n 2 n i 1 
 
 each Z e ft, define an indicator function I as follows: 
 
 Z 
 
 1 if Q„ > K 
 
 I =' z 
 
 otherwise 
 The numbers of Z's for which Q > K is then equal to 
 
 Li 
 
 N A I I 
 Zeft 
 
 Provided .50 is used in (3. 5-6), the expected value of N = \ E{I„} = — |fi| 
 
 Zeft 
 where |fl| denotes the number of elements in the set ft. The histogram of 
 
 B can be predicted from that for B if the behavior around the decile 
 
 n 2 n ± 
 
 points can be quantified. The fact that E{N} = — |ft| indicates that the set of 
 
 points around each decile in the histogram of B should behave in the fol- 
 
 n l 
 lowing way. The deciles of B are approximately C times the deciles of 
 
 n 2 
 
 B since for large Iftl roughly half of the points around each B decile 
 
 n ' ' n 
 
 2k 
 
is expected to change by more than a factor of C with the completion of 
 
 n runs. The other half is expected to change by less than a factor of C. 
 
 Provided some degree ol Independence exists among the entries, the decile at 
 
 n should therefore be approximately c times the decile at n . The final step 
 
 simply uses (3.5-1) to convert the B histogram of values for the Z variables 
 
 n 2 
 in il back to a histogram for the associated X variables. This just amounts 
 
 to multiplying the decile values of B by / 10 since /10a = a and u = u . 
 
 3.^.3 An Example 
 
 As an example, we shall discuss the actual 90 order stopping rule 
 results. After 200 runs, the histograms of Figure 3.5-2 were generated.* 
 As discussed above, B = 3 a / y was chosen as the ordinate since it 
 is a useful measure of the variability of each element in the result 
 set. This figure predicts the variability of results to be expected after 
 different possible stopping points. The diminishing returns for increasing 
 the number of runs is evident from a comparison of the marginal benefit 
 by increasing from 200 to 600 runs to that obtained by increasing from 
 1000 to 2000 runs. This format permits the analyst to decide at which 
 point the marginal benefit no longer justifies the increased cost. Clearly, 
 this requires a non trivial judgement by the analyst. Based on the 
 relatively high cost per iteration in this simulation, 1000 was 
 chosen as the stopping point. 
 
 In an attempt to quantify the accuracy of this stopping rule, comparisons 
 of predicted and actual histograms were made at the 90 order. Results 
 are given in Table 3.5-1. Similar tests were done at the 30 order where 
 actuals were compared with predictions based on only 100 runs, and very 
 small errors were observed. 
 
 *Although similar histograms for the total primary vector, e, and GDO 
 
 were used in the determination of the proper stopping point, for purposes 
 
 of this example, we shall concentrate on figure 3-5-1. 
 
 25 
 
r~ 
 
 «-. 
 
 kj 
 
 ■_; 
 
 £; 
 
 r^ 
 
 <: 
 
 c. 
 
 .^, 
 
 r- 
 
 C 
 
 c 
 
 3 
 
 — 
 
 o 
 
 
 o 
 
 o 
 
 Cj 
 
 Cl 
 
 Ui 
 
 ui 
 
 IM 
 
 UJ 
 
 ui 
 
 u. ' 
 
 u. 
 
 UI 
 
 Ui 
 
 w 
 
 O 
 
 -* 
 
 t>- 
 
 1/ 
 
 o> 
 
 w 
 
 V- 
 
 f^ 
 
 IV 
 
 (J 
 
 <IT 
 
 -o 
 
 o 
 
 J 
 
 »~ 
 
 r- 
 
 -» 
 
 ■Nj 
 
 O 
 
 */■» 
 
 o 
 
 ru 
 
 fl 
 
 r» 
 
 d 
 
 C 
 
 »- 
 
 r^- 
 
 o 
 
 o- 
 
 w-> 
 
 O 
 
 *~ 
 
 *" 
 
 (M 
 
 f"t 
 
 •>f 
 
 v/t 
 
 Cf 
 
 •4 
 
 «- •- o 
 
 1 
 
 l 
 
 -o 
 
 (%j 
 
 r- 
 
 f>» 
 
 >S\ 
 
 K1 
 
 
 ."J 
 
 at 
 
 r~ 
 
 f*. 
 
 •> 
 
 f\i 
 
 » 
 
 KV 
 
 K> 
 
 ft 
 
 IO 
 
 CC 
 
 o 
 
 im 
 
 ■o 
 
 <M 
 
 o- 
 
 o 
 
 •O 
 
 UJ 
 
 r- 
 
 -* 
 
 lO 
 
 "~ 
 
 *" 
 
 (\J 
 
 <M 
 
 »* 
 
 l/^ 
 
 3C 
 
 u~. 
 
 »- «- O 
 
 » <r- «- 
 
 *- I- O 
 
 o» o 
 
 > .- .- 
 
 ui ui Ui 
 
 r- 
 
 «— 
 
 o 
 
 o 
 
 O 
 
 o 
 
 UI 
 
 UI 
 
 UI 
 
 -J 
 
 iSS 
 
 ro 
 
 
 
 •f 
 
 aO 
 
 O •- i- 
 
 o 
 o 
 
 o 
 o 
 
 o 
 
 o 
 
 IA rr «- 
 
 «- «- - C3 
 
 O O 
 
 aj UI 
 
 «- M 
 
 ■J -t 
 
 f- fM 
 
 r> o 
 
 S = 
 
 -O r- t- 
 
 wW IU 
 
 *- oo r^ 
 
 O <- r- 
 
 uu r- r- 
 
 26 
 
3.5.1+ Mathematical Derivations 
 
 (n^ - n - l)n 
 We first demonstrate in this appendix that 73 — ^77 ^ N Q has 
 
 U, - n )(n - l) 
 
 an F/ \ 2 
 
 (n 2 - n ; n ) distribution where Q 4— and then outline the method 
 
 a 
 
 n l 
 for calculating the K of (3.5-6). 
 
 Consider the event R A 
 
 < K ' 
 
 where 
 
 I 7- 
 
 "2 i»l 
 
 n-1 
 
 and y. = z. - y 
 1 1 z. 
 
 1 
 
 f n 2 
 R =< I y, 2 < (n-1) K <? 2 
 i=l X d n i 
 
 Since 
 
 i=l 2 1 
 
 r » 2 
 
 R =(' I y 2 l[(n ? -l) K - (n-1)] a 2 
 j i=n +1 1 
 
 , 2 n-1 
 Dividing "both sides of the inequality by (n -n ) Q ( ) we obtain; 
 
 n 2 
 
 (I y? 
 
 i=n +1 . /(n -n ) 
 
 R = < 
 
 n„ 
 
 ( I 1 y")/^ 
 
 i=l 
 
 L 
 
 2 1 n n 
 
 (n_-l)K - (n.-l) ; n. 
 
 _£ ± I (_ JL_) 
 
 n — n n —1 
 2 J 1 1 
 
 J 
 
 For large samples, y is much closer to the mean than any ^iven sample point, 
 
 2* 
 
 27 
 
z. ,and therefore y. are N(0,o ) and iid. Introducing q. = -- y. in * we obtain 
 
 r 
 
 (I q?)/(n,-nj 
 
 R =< 
 
 i=n +1 
 
 1 x 2 1' 
 
 i i=l 
 
 (n -l)K - (n -1) n. 
 
 — £ , J: ( ±_^ ** 
 
 In -n 
 
 V 1 ' 
 
 where q.,-jN(0,l). Since q.'^ 1 N(0,l) and q. are iid, both the numerator and 
 
 T. 1 1 
 
 v 2 
 
 denominator of the LHS of ** are -k random variables divided by their 
 
 respective degrees of freedom, and the LHS of * and ** have an F, _ n . n ^ 
 
 (n 2 -n 1 -l)n 1 
 distribution [Winkler and Hayes (1970)]. Therefore, -? — - — \ / _^ Q has an 
 
 (n -n in ) distribution, 
 
 QED 
 
 We shall now determine K such that P {Q<K} = .5. By **, this event is 
 
 equivalent to 
 
 ( 
 P ( F 
 
 n ± (n 2 -l)K - (n x -l) jl 
 
 (n 2 -n 1 ;n 1 ) - n -1 
 
 n 2~ n l 
 
 Therefore we simply find ^ such that P <■' F, \ < ■ = .5 and 
 
 V (n 2 - n r n i )_ " 
 
 find K from 
 
 K- Mn 1 -l)(n 2 -n 1 ) + ( j| ^ 
 
 1 
 
 n 2 -l 
 
 28 
 
k.O ANALYSIS OF RESULTS 
 
 The basic results of the simulation will be given for the 90 order 
 1-0 matrix. This includes information on bias relative to published values, 
 variance measures and their relation to error bounds, the sensitivity of the 
 results to uncertainties on the variances of the underlying BEA data and the 
 effects of aggregating to 30 sectors and disaggregating to 101 sectors. As 
 a prelude, we begin by discussing the goodness of fit tests which were 
 required to verify some distributional assumptions inherent in the simulation. 
 
 k.l Goodness of Fit 
 
 The methodology for the goodness of fit tests was developed by Stephens 
 (197°), who describes a test for normality based on the Cramer-von Mises 
 statistic which may be employed when the population mean and standard devia- 
 tion are not known. Stephens' test compares a given sample distribution 
 function to a normal distribution with mean and standard deviation given by 
 the sample mean and sample standard deviation. Included in Stephens' paper is a 
 table of significance levels for the statistic given the hypothesis that the 
 random variable being tested is normal. Thus, a test of normality may be 
 made by calculating the value of Stephens' statistic for a given sample and 
 comparing it to the tabulated values which characterize normal behavior. 
 
 The first series of tests using this method was made to test the normal- 
 ity of the Z random variables defined by averaging every ten consecutive 
 sample points obtained for the entries in the simulation results. In all, 
 270 of these random variables were tested, one for each entry in the electric 
 utility sector row of (i-A) , the total primary energy vector, e, and the 
 total output vector, X. Table U-l shows the upper tail percentage points 
 
 29 
 
calculated by Stephens along with the observed percentages of the Z random 
 variables which fell into the various categories. 
 
 OBSERVED PERCENTAGE 
 
 16.0 
 
 12.22 
 
 6.29 
 
 3.33 
 
 1. 
 
 NORMAL PERCENTAGE 
 
 15.0 
 
 10.0 
 
 5.0 
 
 2.5 
 
 1. 
 
 STEPHENS' STATISTIC 
 
 .091 
 
 .10*4 
 
 .126 
 
 .148 
 
 • 
 
 Table 4-1. A Comparison of Observed and Theoretical Upper Tail Percentage 
 Points for Goodness of Fit Tests on the Random Variables Z. 
 
 For example, Stephens predicts that 10$ of all normal samples will achieve 
 sample statistics larger than .104; we observed 12.2$ above that mark. 
 Even if the 270 random variable being tested are interdependent, the expected 
 value of the observed percentages should equal the theoretical percentages 
 if the normality hypothesis is satisfied. Thus the results are very reas- 
 suring and seem to justify treating the average variables as normal. 
 
 A second series of tests was undertaken to examine the distributional 
 properties of the raw data for the same 270 entries. In the absence of 
 averaging there is little reason to suspect that these random variables are 
 normal. However, the results were surprising in that very many of the 270 
 sample statistics were small and therefore indicate good fit to a normal 
 distribution curve. Those entries that displayed decidedly non-normal be- 
 havior were virtually all unimodal but slightly skewed to the right. It is 
 interesting to conjecture why some entries seem to be roughly normal while 
 others are not; perhaps in the process of inversion some elements of (i-A) 
 get a better mix of elements of the A matrix. At any rate it is useful to 
 know that the entries are all more or less unimodal and symmetric. If such 
 is the case then 3a may be conveniently employed as an error bound on the 
 distance from the mean, p. While Chebychev's inequality guarantees that 
 
 30 
 
y ± 3a contains at least o9% of the total probability in an arbitrary 
 distribution, this percentage rises to 99.7 in the normal case. Pre- 
 sumably the percentage is also high for any random variable whose density 
 function is roughly unimodal and symmetric. For all but one of the entries 
 examined here, at least 99% of the sample points fell within three sample 
 standard deviations of the sample mean. Thus , 3a may be thought of as an 
 approximate bound on deviation from the mean for the entries in the simulation 
 results, even if many of those entries are not very close to being normal. 
 
 k.2 Confidence Intervals 
 
 This section discusses the precision of the sample statistics obtained 
 
 for various simulation results in light of the goodness of fit tests just 
 
 discussed. Because the Z variables are approximately normal, standard 
 
 techniques may be used to derive confidence intervals for the mean and 
 
 variance of a Z variable and hence for the mean and standard deviation of 
 
 the associated entry. After 1000 inversions, a 97-5% upper confidence 
 
 bound a on the standard deviation a of an entry is given by a = a *l.l6. 
 u J u 
 
 Thus, a is a fairly good estimate of a for any given entry. 
 
 The confidence intervals on the sample means are even smaller. In more 
 than 90% of the entries in the inverse the population mean is within 2% of 
 the sample mean with 95% confidence. All the entries of e and X are ac- 
 curate to within 1% with 95% confidence. 
 
 31 
 
^.3 Variability of the Elements in the Result Set 
 
 Histograms of jo/p were prepared in order to show the relative amount 
 
 of variability in the entries of the results set. Three such histograms, one 
 
 for the whole inverse, one fore and one for X, are displayed in Figure k-1. 
 
 For half of the entries of the inverse, 3o/u is less than 20% while 
 
 virtually all the entries of e and X have 3o/p less than 20%. The above 
 
 discussion of confidence intervals suggests that these histograms would not 
 
 change substantially if the sample statistics were replaced by the population 
 
 means and standard deviations. Since these entries are roughly unimodal and 
 
 symmetric, the histograms may then be taken as a good measure of the variabili 
 
 in the entries of the various subsets of the results. The large decrease in 
 
 variability from the elements of the inverse to the elements of X suggests 
 
 that significant error cancellation occurs as linear combinations of many 
 
 1-0 coefficients are computed. 
 
 (3o u + y - p) 
 
 In addition to those discussed above, histograms for 
 
 P 
 
 (3a + p - u) 
 u 
 
 and , where p = published value, were also computed in order 
 
 to relate p to the upper and lower bounds on the uncertainty in an entry. 
 Because y is generally very close to p and because a is only slighter 
 larger than a, these histograms are very similar to the histograms for 
 3o/y except that the values are all slightly larger. 
 
 k.k Bias on Elements of the Result Set 
 
 In standard statistical language, bias is usially defined as the dif- 
 ference between the mean of an estimator and the true value of the quantity 
 to be estimated. We use the term in a fundamentally different way to denote 
 the difference between the mean of the simulation output variables and their 
 
 32 
 
corresponding published values. The mean values of the assumed distributions 
 of each element of the transactions matrix are equal to their respective 
 published values. One important result of the simulation is to determine 
 the bias introduced by normalization and inversion in passing from the trans- 
 actions matrix to (i-A) 
 
 Fig. h-2 details histograms of the ratio of sample mean to published 
 value, u/p, for three important and disjoint subsets of the result set, viz 
 the vectors of total output X and total primary energy intensity e and the 
 entire inverse (i-A) . Three aspects are noteworthy: 
 
 1) Nearly all y cluster within 2% of their published values. 
 
 2) Within this cluster, y tends to have a positive bias more often 
 than a negative one. 
 
 3) Essentially none of the y fall below 9&% of their respective pub- 
 lished values while, especially in the inverse, a small number of 
 y range well above the published value. 
 
 The reason for this positive bias is unclear. The best explanation 
 may be that transactions reported by BEA as zero were assigned a small 
 positive value in the simulation to account for the fact that no trans- 
 action is known to be exactly zero (see Appendix C). The large percentage 
 excess over the published value may result for the same reason, since an 
 inverse element may be affected (percentagewise) quite significantly if 
 its corresponding direct coefficient A. . changes from zero to some finite 
 value . 
 
 k-5. Sensitivity of Simulation Results to Assumptions on Input Uncertainties 
 
 Since the variances assigned to input quantities such as the trans- 
 actions matrix, FD and GDO are only estimates of the true variances, simula- 
 tion results have meaning only if small changes in these assumed variances, 
 
 3i* 
 
do not cause very large changes in simulation outputs. This sensitivity to 
 changes in input variances was investigated by repeating the simulation with 
 the standard deviations of all normal quantities doubled and dispersion 
 factors on lognormal inputs doubled. Three major effects were noted: 
 
 1) The ratio of CI to u, where CI is the length of the 95% confidence 
 interval for the mean, was doubled by the factor of two increase 
 in standard deviation. 
 
 2) The ratio of 3a to y doubled on the average by doubling the input 
 standard deviations. 
 
 3) Increasing the input variability made the biases slightly more 
 negative. This is thought to be the result of increasing simulation 
 sensitivity to the larger elements of the transactions matrix and 
 decreasing relative sensitivity to the smaller elements discussed 
 
 in section k.k. 
 
 Since output uncertainties only doubled with a factor of two increase 
 in input uncertainties, the simulation is probably very stable with regard 
 to assumptions on input variances. 
 
 The absolute magnitudes of these results with doubled input uncertain- 
 ties may be useful in assessing the general viability of 1-0 results applied 
 far beyond the base year (the uncertainty of base year parameters increases 
 over time). Moreover, if institutional factors make it unlikely (as some 
 claim) that government can fairly estimate uncertainty of its own data, then 
 these results show the effect of a 50% underestimate of the actual uncertainty. 
 
 U.6 The Effect of Aggregation 
 
 The effect of aggregating the 90 order model to 30 order was analyzed 
 
 for two reasons : 
 
 l) It was felt that although variances at the 30 order were smaller 
 than those of the 90 order due to the aggregation, more error 
 cancellation should exist at the 90 order where more input elements 
 combined to form elements of X, e and (i-A )"-'-. 
 
 36 
 
2) Since much 1-0 work is done at the 360 order, it is of interest 
 to determine whether the expansion of the simulation to 360 order 
 would likely require more than the 1000 runs used in the 90 order 
 case. 
 
 Aggregation produced effectively no change in the simulation output 
 
 uncertainties: 
 
 1) The ratio of 3a to y remained virtually unchanged by the aggregation. 
 This implies that the cwo effects mentioned above virtually cancel 
 one another. 
 
 2) The already very small biases of Fig. k-2 were made slightly more 
 negative by aggregating to the 30 order. 
 
 3) Since the ratio a /o is a function only of the number of simulation 
 runs, it is unaffected by aggregation. 
 
 These results give no indication that more than 1000 runs would be needed 
 
 in the 360 sector case. 
 
 U.7 Results for the 101 Sector Model 
 
 As discussed in Section 2.2 above, the purpose of the 101 order model 
 is to trade increased base year uncertainty for increased parametric stability 
 over time. The purpose of the 101 order simulation was to measure the 
 increase in base year uncertainty over the 90 order model. Comparison of 
 90 order and 101 order histograms for y /published, CI /y and 3a/y indicates 
 virtually no change in (l-A)~ , GDO and e and a slight increase in 3a /y for 
 the energy related rows of (l-A)~ . In particular, at the 90 order, 95.6$ of 
 the elements of the total primary energy intensities had 3a/y < .15 while at 
 the 101 order, 9h% were less than .15. This indicates a rather low cost in 
 increased stability over time. This low cost is thought to be due to the fact 
 
 3T 
 
that there are many more elements of the transactions matrix which are import- 
 ant to the energy embodied in a particular sector output . Even though 
 the uncertainty of each of these elements is greater in the 101 case, more 
 of them combine so greater error cancellation occurs. 
 
 # 
 
 Consider natural gas to auto manufacturing. Natural gas is sold to perhaps 
 
 eight energy products which in turn are sold to the automobile sector. 
 
 38 
 
APPENDIX A. BASE YEAR UNCERTAINTY ESTIMATES 
 Clark W. Bui lard 
 A.l BE A DATA 
 
 A. 1 . 1 INTRODUCTION 
 
 Input-output data form the basis for most structural analyses of the 
 U.S. economic system. The massive tables of data provide a complete and 
 internally consistent set of linear production functions for all sectors 
 of the economy. But surprisingly, these analytical objectives and appli- 
 cations do not guide the efforts to acquire the data and compile the 
 tables. Actually the input-output tables are constructed as a bridge be- 
 tween the national income and product accounts for selected base years in 
 order to provide a "benchmark GNP" estimate for those years. 
 
 It is important for the analyst using input-output (i-o) data for 
 structural analyses to view the data from this perspective. Since it was 
 not acquired primarily to support structural economic analyses, it places 
 additional burdens on the analyst to verify the data's usefulness and 
 relevance to his particular application. 
 
 Consider the most general type of application, where the analyst 
 wants to predict sector outputs X needed to produce a final bill of 
 goods Y. To do this he premultiplies Y by the Leontief inverse matrix 
 (l-A)~ which is calculated from a matrix of direct coefficients A for a 
 (prior) base year input-output table. The problem the analyst must 
 address is : What is the uncertainty AX on the result X_ given 
 
 Data for 368 sectors are published by the U.S. Department of Commerce (l97^a) 
 
 ** 
 
 For a more detailed discussion of input-output analyses see Leontief (l9Ul). 
 
 39 
 
uncertainty AA in the input-output coefficients? Actually AA may result 
 from 1) base-year measurement error and 2) changes in the actual A since 
 the base year. In this paper we are concerned only with the former. 
 
 Quantitative methods for treating this general error analysis problem 
 are of three types. The first uses the condition number of (i-A) to obtain 
 a bound on the norm of (l_-A) , resulting in an extremely conservative 
 upper bound on parametric error magnification. Ar. expression for true 
 maximum upper bound on (l-A)~ was derived by Secald (1973), and the 
 relative importance of certain parameters to specific applications was 
 determined. Even the true upper bound, however, was quite conservative 
 in that it did not account for the (likely) possibility of error can- 
 cellation. 
 
 This report is limited to presentation of uncertainty estimates on 
 
 1-0 data used in calculating the direct coefficients A. 
 
 A. 1.1.1 Sources of Error 
 
 Uncertainty in the 1-0 coefficients is related directly to several source 
 
 of error in estimating interindustry transactions for the base year. Due 
 to the exhaustive nature of 1-0 data, it originates from a variety of sources 
 ranging from census questionnaires to judgemental guesses. Morganstern (1950 
 has categorized the various sources of error in economic data and most of 
 his observations are relevant here. The total uncertainty on a particular 
 transaction "measurement" will include effects of incomplete census cover- 
 age, reporting errors due to misunderstandings or outright lying, sampling 
 errors inherent in surveys of firms, transcription or key punching errors, 
 
 ho 
 
the possibility that forms are lost, classification errors (matching firms 
 and products to SIC codes), and last but certainly not least, the problem 
 of separating companies from establishments in processing returns from 
 surveys or censuses. 
 
 A. 1.1. 2 Effects of Scale 
 
 The scale of this problem is what makes it unique. Due to the size 
 and complexity of the system being modeled (the U.S. economy) measurments 
 can be taken only at infrequent intervals and at great expense. Moreover, 
 it takes whole institutions to obtain the measurements (e.g. the U.S. Census 
 Bureau) so the user of the data is generally not the one who acquired it. 
 Thus the burden borne routinely by persons who play the roles 
 of data-taker and analyst is now split amoung bureaucracies. Part of that 
 burden — responsibility for estimating parametric uncertainty and its effects 
 on analyses — is sometimes never borne because of the way the roles and 
 responsibilities of the bureaucracies are defined. 
 
 The mission of the Census Bureau is to produce statistics; the Bureau 
 of Economic Analysis (BEA) takes these and others and produces accounting 
 tables supporting a benchmark GNP estimate. The analyst would like to 
 take these statistics and interpret them as observations of a physical sys- 
 tem whose structure he would like to model. The statistics are often 
 published in terms of 5 or 10 significant figures, but none of the hundreds 
 or thousands of persons involved in deriving a statistic are responsible 
 for estimating and documenting its uncertainty. 
 
 kl 
 
A. 1.2 METHOD 
 
 Recognizing that actual measurements of interindustry transactions 
 and other variables are made in the presence of "noise" (error sources), 
 and that frequent measurements are impractical, we must rely on subjective 
 estimates of uncertainty. Such estimates are test obtained at the level 
 of detail at which the measurements are taken, but here too a compromise 
 must be made. A single transaction in an 1-0 table may be the sum of millions 
 , of individual measurements of physical quantities; this report is based 
 on interviews with personnel at BEA, Census, and other agencies near the 
 top of this statistical pyramid. 
 
 A. 1.2.1 Quantities Estimated 
 
 Uncertainty estimates were obtained on the three basic constituents of 
 the interindustry transactions matrix. These were direct allocations, mar- 
 gins on domestic transactions, and transfers* Independently, estimates were 
 obtained for final demands, gross domestic outputs, and imports and exports. 
 
 In the next section, uncertainty estimates will be given for each of 
 these categories of data. 
 
 A. 1.2.2 Degree of Detail 
 
 Within the scope of this study it was possible to consider data inputs 
 to the 1-0 tables at the U8U-sector level of detail in many cases; and at 
 the 368-sector level for the remainder. At the more detailed level, a 
 magnetic tape was available from BEA which included notes fcr various direct 
 allocations indicating the source of the data and the magnitude of the 
 
 « 
 Precise definitions of these terms are given by the U.S. Department of 
 Commerce (l97Ub). 
 
 U2 
 
figure obtained from that source. This tape was scanned for notes identi- 
 fying entries from the Census Bureau or other sources deemed equally ac- 
 curate. If more than 75% of the entry in the 368-order Direct Allocations 
 matrix was from one of these sources, it was assigned the same uncertainty 
 as census data. Estimates of uncertainty for all other data were made at 
 the 368-level of detail as described in the next section. 
 
 A. 1.2. 3 Interviev Techniques 
 
 Many agency personnel seemed well-prepared and sometimes even anxious 
 to assign quantitative estimates of uncertainty to the statistics for which 
 they were responsible. Others were quite reluctant, citing the fact that 
 the "correct" answer was not known and only one measurement had been taken 
 so there was inadequate information on which to base an answer. While this 
 latter group was probably more correct in their assessment of the situation, 
 it should be remembered that such a statement could be used as an "excuse" 
 for covering up error levels that might reflect badly on one's job perfor- 
 mance. In virtually every case, those interviewed responded with a quanti- 
 tative answer to a question of the form "If God appeared and told the 
 correct number to the commander of a firing squad, and if that commander 
 asked you to estimate error bounds for your published figure and threaten- 
 ed to kill you if the correct figure lay outside the bounds ... What would 
 you estimate?" 
 
 During the course of interviews with persons relying on the same data 
 sources, and with persons responsible for producing that source data, I was 
 able to arrive at what I believe to be an internally consistent set of uncer- 
 tainty estimates.' All results presented in this report may be attributed to the 
 
 These sources are Minerals Yearbook , Census of Mineral Industries , Census 
 of Manufactures Table 7A , Census of Transportation , Census of Business , 
 Interstate Commerce Commission, and Civil Aeronautics Board publications. 
 
 k3 
 
author, although footnotes are used to identify the persons whom I inter- 
 viewed to obtain information and impressions. Given the nature of the 
 strong institutional pressures for downward bias in these estimates, I 
 do not expect that the pressures for conservatism that I offered in phras- 
 ing my interview questions provided a significant counteracting force. 
 
 A.I.2.U. Bi 
 
 las 
 
 It is expected that uncertainty estimates obtained from such "top of 
 the pyramid" interviews will be biased downward, since a BEA employee (say) 
 will be reluctant to question the Census Bureau's estimate of the total 
 U.S. steel production unless he has conflicting statistics from somewhere 
 else. Since the Census Bureau has a virtual monopoly on such statistics, 
 the latter situation is impossible; since the BEA employee has barely the 
 resources to do his own job, he cannot begin to duplicate the efforts of 
 the Census Bureau so the former situation never arises either. Simply 
 stated, if one bureaucracy publishes a seven-significant-figure statistic 
 that cost a million dollars to derive, the humble bureaucrat in another 
 agency, with his own problems to worry about, is unlikely to seriously 
 challenge the figure. 
 
 Possible treatments for this problem of bias will be discussed in 
 the last section. 
 
 A. 1.2.5 Effect of Numerical Magnitude 
 
 Development of 1-0 data involves much work within established, or 
 relatively well-known, control totals. For this reason, and since the 
 work is done primarily within an accounting framework, the largest numbers 
 
 « 
 See Morganstern (1950). 
 
 kk 
 
usually receive the most attention and are the best known, and the "residual" 
 between the well -known components and the control total is often distributed 
 among other categories using some kind of estimation algorithm. The only 
 exception to this general "rule" occurs when the figure involved has a signi- 
 ficant impact on the value of GNP; then, though small, the figure may become 
 the subject of further analysis and refinement. 
 
 A. 1.3. UNCERTAINTY ESTIMATES 
 
 In this section, estimates will first be presented at the 368-sector 
 level of detail. This was the level of disaggregation at which most of the 
 persons interviewed were most comfortable in assigning their subjective 
 estimates of uncertainty. 
 
 As indicated earlier, all estimates of upper and lower bounds presented 
 here may be attributed to the author. The discussion and footnotes indicate 
 the source of my impressions and information. 
 
 Estimates of upper and lower bounds are given in two ways. The first 
 is a fraction 6 which denotes symmetric bounds around the published value 
 of + 1006 %. The second, applied in cases where the published value is 
 less well known, is the factor D which when multiplied by the published 
 value gives the upper bound, and whose inverse determines the lower bound. 
 All bounds should be taken to represent a 99- 1% confidence level. 
 
 A. 1.3.1 Direct Allocations 
 
 "Good" Census- gra de entri es. All transactions from one manufacturing 
 sector to another are assigned 6 = .05, as are all other interindustry 
 direct allocations obtained from Census Bureau sources. This figure is 
 
 * 
 
 This information based primarily on interviews with Kenneth Hanson, Richard 
 Chassey, Ruth Runyan, and Patrick Duck of the Census of Manufactures, 
 Industry Division. 
 
 US 
 
based on interviews with Census Bureau personnel who feel their techniques 
 for circumventing problems associated with less than 100$ coverage are 
 well within these limits, and that internal cross-checks minimize report- 
 ing and related errors. The largest source of error here is suspected to 
 
 be classification error; matching products and firms to SIC codes. 
 
 * 
 
 Agriculture sector rows . Based largely on crop reporting surveys; 
 
 estimate 6 = .10 except for certain transactions noted elsewhere 
 (e.g. government final demand). 
 
 Agriculture sector columns . Inputs from real estate, chemicals, and 
 chemical fertilizer mining are known best from surveys and other sources; 
 estimate 6 = .10. Directly allocated inputs from transportation and trade 
 sectors were treated the same as margins, as described in sections 3,*+. All 
 other entries are based at least in part on farm expenditure surveys taken in 
 
 1955; assume D = 2 for all entries greater than 1% of gross domestic out- 
 put for the sector. All smaller nonzero numbers scaled from D = 2 -*■ D = 10 
 as described in Sec. A. 3. 
 
 Federal government purchases . For both defense and non-defense 
 purchases, the following assumptions apply; hew construction inputs are 
 based on a good data source, so assign 6 = .05; maintenance and repair 
 construction is more subject to classification errors, so 6 = .10. 
 All entries between $10 million and $50 million are assigned 5 = . 30 
 unless otherwise specified below. Purchases less than or equal to 
 $10 million are assigned D = 2 -> 10 as discussed in Sec. A. 3. 
 
 *Based primarily on interviews with Jerry Schluter, U.S. Department of 
 Agriculture. 
 
 Based primarily on interviews with Roy Seaton , Bureau of Economic Analysis, 
 
 h6 
 
Defense purchases are generally better known, due to more complete 
 source data. Inputs from manufacturing sectors are assigned 6 = .10 
 if they exceed $50 million. Transportation inputs were derived from 
 outdated formulae that applied poorly to the Southeast Asia situation 
 in 1967 and are assigned 6 = .50. Other non-manufacturing inputs were 
 assigned 6 = .10 if they were above the $50 million threshold. 
 
 Non-defense purchases of inputs from non-manufacturing sectors were 
 less well known, and were assigned D = 3 if they exceeded one percent of 
 total inputs and D=3-*10 if they were smaller, liar.uf acturir.g ir.puts below 
 the $50 million threshold were treated the same. Transportation inputs 
 were assigned 6 = .30. 
 
 State and local government purchases . 7cr health, welfare, education, 
 and sanitation purchases, new construction and real estate inputs are as- 
 signed 6 = .05 since they are obtained from census sources. Together with 
 wages, these inputs account for nearly 75^ of all ir.puts. Other inputs 
 are assigned 6 = .25 if they exceed 1% of total ir.puts, and Z = 1.5 -*■ 10 
 as per Sec. A. 3 if they are equal to or smaller than 1%. 
 
 For public safety purchases, new construction and real estate are 
 assigned 6 = .05. Maintenance construction is known poorly; D = 1.5. 
 Manufactured inputs greater than $2 million are assigned D = 1.5, and 
 smaller inputs D = 1.5 ■* 10. Non-manufactured ir.puts are assigned D = 1. 5 for 
 those greater than $10 million, and D = 1.5 •*■ 10 for the smaller ones. 
 
 Other state and local government purchases are also assigned 5 = .05 
 for new construction and real estate, but also 5 = .05 for maintenance 
 construction since it is primarily highway maintenance which is a Census 
 
 4t 
 
 Based primarily on interviews with John Wealty, Bureau of Economic Analysis, 
 
 1,7 
 
number. Manufactured inputs greater than $5 million are assigned D = 1.5, 
 and smaller figures D ■ 1.5 ■+• 10 ae per Sec. A. 3. Non-manufactured inputs 
 
 greater than $50 million are assigned D = 2 and D = 2 -*■ 10 for smaller inputs 
 
 # 
 Imports and exports . Trade data for commodities (BEA sectors 1.00 - 
 
 6U.00) are obtained from Census sources and are assigned 6 = .05. Trans- 
 portation and wholesale and retail trade data, including margins, were assign* 
 6 = .25. Data on other items (services, etc.) involved in international 
 trade were assigned D = 2, since they were obtained from balance of pay- 
 ments sample data. Small entries at the 368-sector level of detail, repre- 
 senting less than 1% of gross imports or exports were assigned D = 2 ■* 10 
 as per Sec. A. 3. 
 
 Inventory change . These figures are in general the least accurate 
 of all final demand entries, and were assigned 6 = .20 for manufactured 
 goods and 6 = .40 elsewhere. 
 
 "All other" direct allocations . Within the scope of this study it was 
 impossible to identify those responsible for most entries in the input- 
 output tables. Having taken care of most entries through interviews des- 
 cribed above, the remainder were handled as a group. The algorithm was 
 designed to assign very tight tolerances to any transaction comprising a 
 high percentage of total outputs or inputs, and to any sector's output 
 which "by definition" had to be assigned to a particular cell. For example, 
 the algorithm had to assign a very tight tolerance to sales from new resi- 
 dential construction to gross private capital formation, so it would be 
 compatible with the tolerance assigned to that sector's gross domestic out- 
 put. There are numerous other instances where census data might identify 
 
 
 
 Based primarily on interviews with Robert Mangen, Bureau of Economic Analysis. 
 
 kQ ■ 
 
 
sales of "butter to food processors or bakers, and the remainder is attri- 
 buted to personal consumption expenditures. On the other hand, very small- 
 magnitude transactions were assigned high uncertainty for the reasons dis- 
 cussed earlier. 
 
 The algorithm defined two fractions for each direct allocation: 
 an input fraction, by normalizing with respect to the gross domestic 
 output of the consuming sector; and an output fraction, by normalizing 
 with respect to the gross domestic output of the producing sector. The 
 algorithm proceeds with these tests in the following order, and assigning 
 6 or D when the first condition is satisfied: if both fractions exceed 
 •95 then 6 = .01, if only one exceeds .95 then 5 = .02; if both exceed 
 .80, 6 = .05, if only one exceeds .80, 6 = .10; if either fraction ex- 
 ceeds .05, then 6 = .20; if either exceeds .01, then Z - 1.5. If 
 both are smaller than .01 it assigns D = 2 -*■ 10 as per Sec. A. 3. 
 
 * 
 A. 1.3. 2 Gross Domestic Output 
 
 These figures are the best known because they are from the Census 
 
 or other equally reliable sources (e.g., IRS) and are assigned 5 = .01. 
 
 The largest errors here probably stem from classification problems and 
 
 possible confusion between company and establishment-based data. 
 
 A. 1.3. 3 Transfers . 
 
 If both the row and column sectors were manufacturing sectors , the 
 
 Based primarily on interviews with Gene Roberts ana Phil Ritz , Bureau oi 
 Economic Analysis, and with Kenneth Hanson, Census of Manufactures Industry 
 Division. 
 
 #* 
 
 Based primarily on interviews with Kenneth Hanson, Census of Manufactures 
 
 Industry Division. 
 
 U9 
 
source of this data was the Census Bureau, cut the accuracy was less than 
 that of direct allocations; assign 6 = .20. All other transfers were 
 assigned upper and lower bounds in the same manner as the corresponding 
 cell in the direct allocations matrix. 
 
 A. 1.3.*+ Margins 
 
 Transportation margins, Toy product type and mode, are obtained as 
 
 totals and then prorated proportional to producers' prices across all pur- 
 chasers of that commodity. Then margins in each .input are summed for 
 each purchaser and added to the directly allocated inputs. For all 
 transport modes, 6 = .25 was assigned to the margins, Wholesale and 
 retail trade margins may be expected to be more variable, and are some- 
 times computed as percentage markups over the already estimated trans- 
 port-margins. Therefore they are assigned 6 - .35- 
 
 A.l.U. CONCLUSIONS, APPLICATIONS, AND LIMITATIONS 
 
 Earlier work using maximum-upper-bound analyses had shown the dangers 
 that might be encountered using results of input-output analyses. There- 
 fore, these estimates of uncertainty on the actual data were needed to check 
 the maximum error bounds on the particular results we were interested in 
 using (e.g. , elements of the energy sector rows of the 1967 Leontief inverse 
 matrix) . It soon became evident that the magnitude of the uncertainties in 
 
 *See for example the results presented by Bullard and Sebald (1975). 
 
 50 
 
the parameter estimation process that our max' mum upper bound analysis would 
 yield unsatisfactory results. 
 
 The above information is given to further i.lluminat'e the context in 
 which these uncertainty estimates were made, nnd hopefully will dis- 
 courage inappropriate applications of the results. 
 
 Finally, I repeat that the uncertainty estimates presented here 
 are my own. I have listed many of the persons whom I interviewed, but 
 they have not endorsed my interpretations of those interviews. If the 
 absolute levels of the estimates are widely disputed (and I expect they 
 will be) perhaps at least the relative levels will be accepted. On 
 
 this basis we have performed stochastic error analyses on the 1°67 U.S. 
 input-output model for several cases; including doubling error margins 
 presented here, to determine the sensitivity of the results to systematic 
 bias in the estimates. 
 
 51 
 
A. 2 DIRECT ENERGY ALLOCATIONS 
 
 Knecht (1975) estimated error tolerances on all physical-unit energy- 
 transactions. These are coded in Table A. 2-2, at the 90 sector level of 
 detail, and in Tables A. 2-3 and A.2-U at the 101-sector level, and the" codes ai 
 explained in Table A. 2-1 below. 
 
 Table A. 2-1 
 ENERGY TRANSACTION TOLERANCE CODES 
 
 Code 
 
 00 y=0 and 3a=10 11 Btu) 
 
 01, 02,09,13 .05 
 
 0U, Ul,l6, 18, 19,20,2^,28 .10 
 
 03,05,06,29,30 .15 
 
 07, 12, Ik, 15, 17, 22, 23, 26, 27 .20 
 
 25 .25 
 
 08,10,11 .30 
 
 10 .35 
 
 * Note that instead of the 368-sector level of aggregation, the results pre- 
 sented here are consistent with the slightly aggregated 357-sector breakdown 
 described by Bullard & Herendeen (1975). Dummy sectors consuming no energy 
 have been deleted and public and private sectors producing the same primary 
 product have been combined. 
 
 52 
 
TABLE A. 2-2 
 
 TOLERANCE CODES FOR DIRECT ENERGY USE DATA (90 SECTOR) 
 
 Sector 
 
 Number 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 10 
 11 
 12 
 13 
 14 
 15 
 16 
 17 
 18 
 19 
 20 
 21 
 22 
 23 
 2k 
 25 
 26 
 27 
 28 
 29 
 30 
 31 
 32 
 33 
 34 
 35 
 36 
 37 
 38 
 39 
 40 
 41 
 42 
 43 
 44 
 45 
 46 
 47 
 48 
 49 
 50 
 
 Sector Name 
 
 COAL MINING 
 CRUDE PETR0LEU1/ GAS 
 REF'D PETROLEUM PROD'S 
 ELECTRIC UTILITIES 
 NATURAL GAS UTILITIES 
 LIVESTOCK.. .PRODUCTS 
 OTHER AGRIC'L PRODUCTS 
 FORESTRY AND FISHERY.. 
 AG./ FOR'Y. ..SFRVICES 
 IRON.. .OPES MINING 
 NONFERROUS ORES MINIMS 
 STONE AND CLAY MINING. 
 CHEMICALS^ ETC. MINING 
 NEW CONSTRUCTION 
 MAINT. AND REPAIR CON. 
 ORDNANCE AND ACCESSOR. 
 FOOD AND KINDRED PROD. 
 TOBACCO MANUFACTURING 
 FABRIC. ..THREAD MILLS 
 MISC. TEXTILE. ..FLOOR. 
 APPAREL 
 
 MISC. FA3. TEXTILE PRO 
 LUMBER. ..PROD'S/ EXCEP 
 WOODEN CONTAINERS 
 HOUSEHOLD FURNITURE 
 
 OTHER FURNITURE AND 
 
 PAPER AND.. .EXCEPT... 
 
 PAPERB'D CONTAINERS 
 
 PRINTING AND PU3LISH'G 
 CHEMICALS AND. ..PROD'S 
 
 PLASTICS AND MATER'S 
 
 DRUGS /...PREPARATIONS 
 
 PAINTS AND PRODUCTS 
 
 PAVING MIXTURES AND... 
 ASPHALT FELTS AND COAT 
 RUBBER AND. ..PRODUCTS 
 LEATHER TANNING AND... 
 FOOTWEAR AND. ..PROD'S 
 GLASS AND GLASS PROD'S 
 STONE AND CLAY PROD'S 
 PRIM. IRON AND STEEL.. 
 PRIM. NONFERROUS METAL 
 METAL CONTAINERS 
 
 HEAT./ PLUMti PROD'S 
 
 SCREW MACH. PROD'S/ 
 
 OTHER FAO. METAL PROD. 
 ENGINES AND TURH1NES 
 FARM MACHINERY 
 
 CONSTRUCTION/ EQUIP. 
 
 MAT. HANDLING EQUIP. 
 
 Coal 
 
 Energy Supplies 
 Crude Oil Electric 
 
 Gas 
 
 02 
 
 00 
 
 02 
 
 02 
 
 02 
 
 11 
 
 02 
 
 02 
 
 02 
 
 02 
 
 01 
 
 01 
 
 01 
 
 01 
 
 01 
 
 01 
 
 00 
 
 01 
 
 01 
 
 01 
 
 11 
 
 01 
 
 11 
 
 11 
 
 01 
 
 11 
 
 00 
 
 08 
 
 08 
 
 11 
 
 11 
 
 00 
 
 08 
 
 08 
 
 11 
 
 11 
 
 00 
 
 11 
 
 11 
 
 11 
 
 11 
 
 00 
 
 11 
 
 11 
 
 11 
 
 11 
 
 00 
 
 02 
 
 02 
 
 02 
 
 11 
 
 00 
 
 02 
 
 02 
 
 02 
 
 02 
 
 00 
 
 02 
 
 02 
 
 02 
 
 11 
 
 00 
 
 02 
 
 02 
 
 02 
 
 03 
 
 00 
 
 11 
 
 11 
 
 11 
 
 03 
 
 00 
 
 11 
 
 11 
 
 11 
 
 03 
 
 00 
 
 05 
 
 03 
 
 03 
 
 02 
 
 00 
 
 04 
 
 02 
 
 02 
 
 02 
 
 00 
 
 04 
 
 02 
 
 02 
 
 02 
 
 00 
 
 04 
 
 02 
 
 02 
 
 03 
 
 00 
 
 05 
 
 11 
 
 03 
 
 03 
 
 00 
 
 05 
 
 03 
 
 03 
 
 11 
 
 00 
 
 11 
 
 11 
 
 11 
 
 03 
 
 00 
 
 05 
 
 03 
 
 03 
 
 11 
 
 uo 
 
 05 
 
 03 
 
 11 
 
 03 
 
 00 
 
 05 
 
 03 
 
 03 
 
 03 
 
 00 
 
 05 
 
 11 
 
 03 
 
 02 
 
 00 
 
 04 
 
 02 
 
 02 
 
 02 
 
 00 
 
 04 
 
 02 
 
 02 
 
 03 
 
 00 
 
 05 
 
 03 
 
 03 
 
 02 
 
 02 
 
 04 
 
 02 
 
 02 
 
 02 
 
 00 
 
 04 
 
 02 
 
 02 
 
 02 
 
 00 
 
 04 
 
 02 
 
 02 
 
 02 
 
 00 
 
 04 
 
 02 
 
 02 
 
 11 
 
 00 
 
 06 
 
 02 
 
 02 
 
 02 
 
 00 
 
 06 
 
 02 
 
 02 
 
 02 
 
 00 
 
 04 
 
 02 
 
 02 
 
 03 
 
 00 
 
 05 
 
 11 
 
 11 
 
 05 
 
 00 
 
 05 
 
 03 
 
 03 
 
 02 
 
 00 
 
 04 
 
 02 
 
 02 
 
 02 
 
 00 
 
 04 
 
 02 
 
 02 
 
 02 
 
 00 
 
 04 
 
 02 . 
 
 02 
 
 02 
 
 00 
 
 04 
 
 02 
 
 02 
 
 02 
 
 00 
 
 06 
 
 02 
 
 02 
 
 03 
 
 00 
 
 05 
 
 03 
 
 03 
 
 02 
 
 00 
 
 04 
 
 02 
 
 02 
 
 03 
 
 00 
 
 07 
 
 03 
 
 03 
 
 02 . 
 
 00 
 
 04 
 
 02 
 
 02 
 
 02 
 
 00 
 
 04 
 
 02 
 
 02 
 
 03 
 
 00 
 
 05 
 
 03 
 
 03 
 
 03 
 
 00 
 
 05 
 
 11 
 
 03 
 
 53 
 
TABLE A. 2-2 (continued) 
 
 Sector 
 Number 
 
 51 
 52 
 53 
 54 
 55 
 56 
 57 
 58 
 59 
 60 
 61 
 62 
 63 
 64 
 65 
 66 
 67 
 68 
 69 
 70 
 71 
 72 
 73 
 74 
 75 
 76 
 77 
 78 
 79 
 80 
 81 
 82 
 83 
 84 
 85 
 86 
 87 
 88 
 89 
 90 
 91 
 92 
 93 
 94 
 95 
 96 
 97 
 98 
 99 
 100 
 
 Sector Name 
 
 Energy Supplies 
 
 METALS 
 
 SPEC. 
 
 GEN. I 
 
 M A C M J N 
 
 OFF., 
 
 SERV. 
 
 ELEC. 
 
 HOUSt H 
 
 ELEC. 
 
 RADIO, 
 
 ELEC. 
 
 MISC. 
 
 MOTOR 
 
 A1RCRA 
 
 OTHER 
 
 PROFES 
 
 OPICAL 
 
 MISC. 
 
 RA1LRO 
 
 ...HIG 
 
 MOTOR 
 
 WATER 
 
 AIR TR 
 
 PIPE L 
 
 TRANSP 
 
 COM'. MS 
 
 RADIO 
 
 WATER 
 
 WHOLES 
 
 FINANC 
 
 REAL E 
 
 HOTELS 
 
 BUSINE 
 
 AUTO. 
 
 AMUSEM 
 
 MED./ 
 
 FED. G 
 
 STATE 
 
 BUS. T 
 
 OFFICE 
 
 PERS. 
 
 GROSS. 
 
 NET IN 
 
 NET EX 
 
 FED. G 
 
 FED. G 
 
 STATE. 
 
 STATE. 
 
 STATF. 
 
 STATE. 
 
 0RK1N 
 
 INDUS 
 
 ND'JST 
 
 E ShO 
 
 C'OMP' 
 
 1ND. 
 
 TRANS 
 
 OLD A 
 
 LI'jHT 
 
 TV, 
 CO*!PO 
 ELEC. 
 VEHIC 
 FT AN 
 TRANS 
 SIONA 
 . ..EQ 
 M A N U F 
 ADS A 
 HwAY 
 FREIG 
 TRANS 
 ANSPO 
 INE T 
 ORTAI 
 
 EXCE 
 AND T 
 AND S 
 ALE A 
 E AND 
 STATE 
 
 AND. 
 SS SE 
 REPAI 
 ENTS 
 ED. S 
 OV'T 
 AND L 
 RAV.. 
 
 SUPP 
 CONSU 
 ..CAP 
 V E N T 
 PORTS 
 OV'T. 
 OV'T. 
 . .GOV 
 . .HFA 
 . .GOV 
 . .GOV 
 
 'j . . . E 
 TRY.. 
 RIAL. 
 P PRO 
 
 G M 
 
 M A C H 1 
 
 ...AP 
 
 PPLIA 
 
 •G... 
 
 COM. 
 
 NENTS 
 
 ..SUP 
 
 HLES. 
 
 D PAR 
 
 . E9IJ 
 
 L...S 
 
 UIP. 
 
 ACTUR 
 
 ND 
 
 PASS . 
 HT TR 
 PORTA 
 RTATI 
 RANSP 
 ON SE 
 PT RA 
 V 3R0 
 AN. S 
 ND RE 
 INSU 
 AND 
 ..EXC 
 "VICE 
 R AND 
 
 QUIP'T 
 
 .EQUIP 
 
 . .EQ'T 
 
 DUCTS 
 
 mCHINE 
 
 NES 
 
 PARAT. 
 
 NCES 
 
 EQUIP. 
 
 EQUIP. 
 
 PLIES 
 
 ..EQ'T 
 
 TS 
 
 IP*1ENT 
 
 UPPLIE 
 
 AND. .. 
 
 1NG 
 
 SERV'S 
 
 TRAN. 
 ANS. .. 
 T10N 
 ON 
 
 ORTA'N 
 RVICES 
 DID... 
 ADCAST 
 ERV 'S 
 TAIL.. 
 RANCE 
 RENTAL 
 EPT... 
 S 
 
 SERV. 
 
 ERV'S AND.. 
 
 ENTERPRISES 
 
 OCAL...EN'S 
 
 .AND GIFTS 
 
 LIES 
 
 MP'N EXPEN. 
 
 . FORMATION 
 
 RY CHANGE 
 
 ..DEFENSE 
 ..OTHER 
 'T...EDUC 'N 
 
 LTH, SAN. 
 
 'T.. .SAFETY 
 •T OTHER 
 
 sal 
 
 Crude 
 
 Oil 
 
 Electric 
 
 Gas 
 
 U2 
 
 00 
 
 04 
 
 02 
 
 02 
 
 03 
 
 00 
 
 05 
 
 11 
 
 03 
 
 02 
 
 00 
 
 04 
 
 02 
 
 02 
 
 03 
 
 00 
 
 05 
 
 03 
 
 03 
 
 11 
 
 00 
 
 04 
 
 02 
 
 02 
 
 03 
 
 00 
 
 05 
 
 11 
 
 03 
 
 03 
 
 00 
 
 05 
 
 03 
 
 03 
 
 03 
 
 00 
 
 05 
 
 11 
 
 03 
 
 03 
 
 00 
 
 05 
 
 03 
 
 03 
 
 03 
 
 00 
 
 05 
 
 03 
 
 03 
 
 02 
 
 00 
 
 04 
 
 02 
 
 02 
 
 03 
 
 00 
 
 05 
 
 11 
 
 11 
 
 02 
 
 00 
 
 04 
 
 02 
 
 02 
 
 02 
 
 00 
 
 04 
 
 02 
 
 02 
 
 0? 
 
 00 
 
 04 
 
 02 
 
 02 
 
 03 
 
 00 
 
 05 
 
 11 
 
 11 
 
 03 
 
 00 
 
 04 
 
 02 
 
 02 
 
 03 
 
 00 
 
 05 
 
 03 
 
 11 
 
 09 
 
 00 
 
 09 
 
 09 
 
 11 
 
 11 
 
 00 
 
 09 
 
 09 
 
 11 
 
 11 
 
 00 
 
 09 
 
 11 
 
 11 
 
 0? 
 
 00 
 
 09 
 
 11 
 
 11 
 
 11 
 
 00 
 
 09 
 
 11 
 
 11 
 
 11 
 
 00 
 
 11 
 
 11 
 
 09 
 
 11 
 
 00 
 
 00 
 
 11 
 
 11 
 
 11 
 
 00 
 
 11 
 
 11 
 
 11 
 
 11 
 
 00 
 
 11 
 
 11 
 
 11 
 
 11 
 
 00 
 
 11 
 
 11 
 
 11 
 
 11 
 
 00 
 
 11 
 
 11 
 
 11 
 
 11 
 
 00 
 
 11 
 
 12 
 
 11 
 
 no 
 
 00 
 
 11 
 
 11 
 
 11 
 
 11 
 
 00 
 
 11 
 
 11 
 
 51 
 
 11 
 
 00 
 
 11 
 
 12 
 
 11 
 
 n 
 
 00 
 
 11 
 
 12 
 
 11 
 
 11 
 
 00 
 
 11 
 
 12 
 
 11 
 
 11 
 
 00 
 
 11 
 
 11 
 
 11 
 
 11 
 
 00 
 
 11 
 
 12 
 
 11 
 
 11 
 
 00 
 
 11 
 
 12 
 
 11 
 
 00 
 
 00 
 
 00 
 
 00 
 
 00 
 
 03 
 
 00 
 
 00 
 
 00 
 
 00 
 
 11 
 
 00 
 
 01 
 
 12 
 
 01 
 
 CO 
 
 00 
 
 00 
 
 00 
 
 00 
 
 01 
 
 01 
 
 01 
 
 00 
 
 01 
 
 01 
 
 01 
 
 01 
 
 01 
 
 01 
 
 11 
 
 00 
 
 11 
 
 11 
 
 11 
 
 11 
 
 00 
 
 11 
 
 11 
 
 11 
 
 11 
 
 00 
 
 11 
 
 11 
 
 11 
 
 11 
 
 00 
 
 11 
 
 12 
 
 11 
 
 11 . 
 
 00 
 
 11 
 
 12 
 
 11 
 
 11 
 
 00 
 
 11 
 
 12 
 
 11 
 
 5U 
 
TABLE A. 2-3 
 
 TOLERANCE CODES FOR ENERGY END USES DATA (101 SECTOR) 
 
 
 
 
 
 
 Misc. 
 
 
 
 
 Misc. 
 Elec. 
 
 Sector 
 
 i 
 
 
 Feed- 
 
 Mot. 
 
 Ther. 
 
 Water 
 
 Space 
 
 Air 
 
 Pow. 
 
 Number 
 
 Sector Name 
 
 Coke 
 
 Stocks 
 
 Pow. 
 
 Users 
 
 Heat 
 
 Heat 
 
 Cond 
 
 • Uses 
 
 1 
 2 
 
 COAL MINING 
 
 00 
 
 14 
 
 14 
 
 16 
 
 15 
 
 00 
 
 00 
 
 18 
 
 CRUDE PETROLEUM/- GAS 
 
 00 
 
 14 
 
 14 
 
 16 
 
 15 
 
 00 
 
 00 
 
 18 
 
 3 
 
 GASIFIED COAL 
 
 00 
 
 21 
 
 '21 
 
 21 
 
 21 
 
 21 
 
 21 
 
 21 
 
 k 
 
 RtF'D PETROLEUM PROD'S 
 
 00 
 
 13 
 
 14 
 
 19 
 
 15 
 
 14 
 
 1 7 
 
 20 
 
 5 
 
 NATURAL GAS UTILITIES 
 
 00 
 
 14 
 
 14 
 
 00 
 
 14 
 
 22 
 
 14 
 
 23 
 
 6 
 
 FOSSIL ELECTRIC UTIL'S 
 
 00 
 
 14 
 
 14 
 
 30 
 
 14 
 
 22 
 
 14 
 
 23 
 
 7 
 
 NUCLEAR ELEC. UTIL'S 
 
 00 
 
 14 
 
 14 
 
 00 
 
 14 
 
 22 
 
 14 
 
 23 
 
 8 
 
 RENEWA3LE ELEC. UTIL'S 
 
 00 
 
 14 
 
 14 
 
 00 
 
 14 
 
 22 
 
 14 
 
 23 
 
 9 
 
 ORt-REDUC. FEEDSTOCKS 
 
 00 
 
 00 
 
 00 
 
 DO 
 
 00 
 
 00 
 
 00 
 
 00 
 
 10 
 
 CHEMICAL FEEDSTOCKS 
 
 00 
 
 00 
 
 DO 
 
 30 
 
 00 
 
 00 
 
 00 
 
 00 
 
 11 
 
 MOTIVE POWER 
 
 00 
 
 00 
 
 00 
 
 00 
 
 00 
 
 00 
 
 00 
 
 00 
 
 12 
 
 MISC. THERMAL USES 
 
 00 
 
 00 
 
 00 
 
 30 
 
 DO 
 
 00 
 
 UO 
 
 00 
 
 13 
 
 WATER HEAT 
 
 00 
 
 00 
 
 00 
 
 30 
 
 30 
 
 00 
 
 00 
 
 00 
 
 Ik 
 
 SPACE HEAT 
 
 00 
 
 00 
 
 00 
 
 30 
 
 00 
 
 00 
 
 00 
 
 00 
 
 15 
 
 AIR-CONDITIONING 
 
 00 
 
 00 
 
 00 
 
 00 
 
 00 
 
 00 
 
 00 
 
 00 
 
 16 
 
 MISC. ELEC. POWER USES 
 
 00 
 
 00 
 
 00 
 
 00 
 
 DO 
 
 00 
 
 00 
 
 00 
 
 17 
 
 LIVESTOCK PRODUCTS 
 
 CO 
 
 u 
 
 14 
 
 00 
 
 14 
 
 22 
 
 00 
 
 23 
 
 18 
 
 OTHER AGRIC'L PRODUCTS 
 
 00 
 
 13 
 
 13 
 
 30 
 
 14 
 
 22 
 
 00 
 
 23 
 
 19 
 
 FORTSTRY AND FISHERY.. 
 
 00 
 
 14 
 
 14 
 
 00 
 
 14 
 
 22 
 
 00 
 
 23 
 
 20 
 
 AG.*- FOR*Y. -.SERVICES 
 
 00 
 
 14 
 
 14 
 
 00 
 
 14 
 
 22 
 
 00 
 
 23 
 
 21 
 
 IRON. ..ORES MINING 
 
 00 
 
 n 
 
 14 
 
 16 
 
 15 
 
 00 
 
 00 
 
 18 
 
 22 
 
 NONFfcRROUS ORES MINING 
 
 00 
 
 14 
 
 14 
 
 16 
 
 15 
 
 00 
 
 00 
 
 18 
 
 23 
 
 STONE AND CLAY MINING. 
 
 00 
 
 14 
 
 14 
 
 16 
 
 15 
 
 00 
 
 00 
 
 18 
 
 2U 
 
 CHEMICALS, ETC. MINING 
 
 00 
 
 14 
 
 14 
 
 16 
 
 15 
 
 00 
 
 00 
 
 18 
 
 25 
 
 NEW CONSTRUCTION 
 
 00 
 
 24 
 
 14 
 
 25 
 
 25 
 
 25 
 
 00 
 
 23 
 
 26 
 
 MAINT. AND REPAIR CON. 
 
 00 
 
 24 
 
 14 
 
 25 
 
 25 
 
 00 
 
 00 
 
 23 
 
 27 
 
 ORDNANCE AND ACCESSOR. 
 
 00 
 
 14 
 
 14 
 
 29 
 
 15 
 
 14 
 
 17 
 
 30 
 
 28 
 
 FOOD AND KINDRED PROD. 
 
 13 
 
 14 
 
 14 
 
 19 
 
 15 
 
 14 
 
 17 
 
 20 
 
 29 
 
 TOBACCO MANUFACTURING 
 
 00 
 
 14 
 
 14 
 
 19 
 
 15 
 
 14 
 
 17 
 
 20 
 
 30 
 
 FA3RIC THREAD MILLS 
 
 00 
 
 14 
 
 14 
 
 19 
 
 15 
 
 14 
 
 17 
 
 20 
 
 31 
 
 MISC. TEXTILE.. .FLOOR. 
 
 00 
 
 14 
 
 14 
 
 29 
 
 15 
 
 14 
 
 17 
 
 30 
 
 32 
 
 APPAREL 
 
 00 
 
 14 
 
 14 
 
 29 
 
 15 
 
 14 
 
 17 
 
 30 
 
 33 
 
 MISC. FAB. TEXTILE PRO 
 
 00 
 
 14 
 
 14 
 
 29 
 
 15 . 
 
 14 
 
 17 
 
 30 
 
 3U 
 
 LUMBER. ..PROD'S, EXCEP 
 
 00 
 
 14 
 
 14 
 
 29 
 
 15 
 
 14 
 
 17 
 
 30 
 
 35 
 
 WOODEN CONTAINERS 
 
 00 
 
 14 
 
 14 
 
 29 
 
 15 
 
 14 
 
 00 
 
 30 
 
 36 
 
 HOUSEHOLD FURNITURE 
 
 00 
 
 14 
 
 14 
 
 29 
 
 15 
 
 14 
 
 00 
 
 30 
 
 37 
 
 OTHER FURNITURE AND 
 
 00 
 
 14 
 
 14 
 
 29 
 
 15 
 
 14 
 
 00 
 
 30 
 
 38 
 
 PAPER AND.. .EXCEPT 
 
 00 
 
 13 
 
 14 
 
 19 
 
 15 
 
 14 
 
 17 
 
 20 
 
 39 
 
 PAPERB'D CONTAINERS... 
 
 00 
 
 14 
 
 14 
 
 19 
 
 15 
 
 14 
 
 17 
 
 20 
 
 UO 
 Ul 
 U2 
 »»3 
 kk 
 h5 
 k6 
 hi 
 
 PRINTING AND PUBLISH'G 
 
 00 
 
 14 
 
 14 
 
 29 
 
 15 
 
 14 
 
 17 
 
 30 
 
 CHEMICALS AND PROD'S 
 
 00 
 
 13 
 
 14 
 
 19 
 
 15 
 
 14 
 
 17 
 
 20 
 
 PLASTICS AND. . .MATER'S 
 
 00 
 
 13 
 
 14 
 
 19 
 
 15 
 
 14 
 
 17 
 
 20 
 
 DRUGS, PREPARATIONS 
 
 00 
 
 14 
 
 14 
 
 19 
 
 15 
 
 14 
 
 17 
 
 20 
 
 PAINTS AND PRODUCTS 
 
 00 
 
 13 
 
 14 
 
 19 
 
 15 
 
 14 
 
 17 
 
 20 
 
 PAVING MIXTURES AND... 
 
 00 
 
 13 
 
 14 
 
 19 
 
 15 
 
 14 
 
 00 
 
 20 
 
 ASPHALT FELTS AND COAT 
 
 00 
 
 13 
 
 14 
 
 19 
 
 15 
 
 14 
 
 00 
 
 20 
 
 RUBBER AND PRODUCTS 
 
 00 
 
 14 
 
 14 
 
 19 
 
 15 
 
 14 
 
 17 
 
 20 
 
 J*8 
 
 LEATHER TANNING AND 
 
 00 
 
 14 
 
 14 
 
 29 
 
 15 
 
 14 
 
 00 
 
 30 
 
 k9 
 
 FOOTWEAR AND PROD'S 
 
 00 
 
 14 
 
 14 
 
 29 
 
 15 
 
 14 
 
 00 
 
 30 
 
 50 
 
 GLASS AND GLASS PROD'S 
 
 00 
 
 14 
 
 14 
 
 19 
 
 15 
 
 14 
 
 17 
 
 20 
 
 51 
 
 STONE AND CLAY PROD'S 
 
 13 
 
 14 
 
 14 
 
 19 
 
 15 
 
 14 
 
 17 
 
 20 
 
 52 
 
 PRIM. IRON AND STEEL.. 
 
 13 
 
 14 
 
 14 
 
 19 
 
 15 
 
 14 
 
 17 
 
 20 
 
 53 
 
 PRIM. NONFERROUS METAL 
 
 13 
 
 14 
 
 14 
 
 19 
 
 15 
 
 14 
 
 17 
 
 20 
 
 55 
 
TABLE A. 2- 3 (continued) 
 
 
 
 
 
 
 
 
 
 
 Misc 
 
 
 
 
 
 
 Misc. 
 
 
 
 
 Elec 
 
 Sector 
 
 
 
 Feed- 
 
 Mot. 
 
 Ther. 
 
 Hater 
 
 Space 
 
 Air- 
 
 Pow. 
 
 Number 
 
 Sector Name 
 
 Coke 
 
 Stocks 
 
 Pow. 
 
 Uses. 
 
 Heat 
 
 Heat 
 
 Cond. 
 
 Uses 
 
 5k 
 
 METAL CONTAINERS 
 
 00 
 
 14 
 
 14 
 
 19 
 
 15 
 
 14 
 
 OC 
 
 20 
 
 55 
 
 HEAT./ PLUMB. . .PROD'S 
 
 13 
 
 14 
 
 14 
 
 29 
 
 15 
 
 14 
 
 17 
 
 30 
 
 56 
 
 SCREW MACH. PROD 'S/ . . . 
 
 CO 
 
 14 
 
 14 
 
 19 
 
 15 
 
 14 
 
 17 
 
 20 
 
 57 
 
 OTHER FAB. METAL TROD. 
 
 13 
 
 14 
 
 14 
 
 29 
 
 15 
 
 14 
 
 17 
 
 30 
 
 58 
 
 ENGINES AND TURBINES 
 
 13 
 
 14 
 
 14 
 
 19 
 
 15 
 
 14 
 
 17 
 
 20 
 
 59 
 
 FARM MACHINERY 
 
 13 
 
 14 
 
 14 
 
 19 
 
 15 
 
 14 
 
 1 7 
 
 20 
 
 60 
 
 CONSTRUCTION, ...EQUIP. 
 
 13 
 
 14 
 
 14 
 
 29 
 
 15 
 
 14 
 
 17 
 
 30 
 
 61 
 
 MAT. HANDLING E0U1P. 
 
 00 
 
 14 
 
 14 
 
 29 
 
 15 
 
 14 
 
 17 
 
 30 
 
 62 
 
 METAL WORKING EQUIP' T 
 
 00 
 
 14 
 
 14 
 
 19 
 
 15 
 
 14 
 
 17 
 
 20 
 
 63 
 
 SPEC. INDUSTRY EQUIP 
 
 13 
 
 14 
 
 14 
 
 29 
 
 15 
 
 14 
 
 1 7 
 
 30 
 
 6k 
 
 GEN. INDUSTRIAL. ..EQ'T 
 
 13 
 
 14 
 
 14 
 
 19 
 
 15 
 
 14 
 
 17 
 
 20 
 
 65 
 
 MACHINE SHOP PRODUCTS 
 
 13 
 
 14 
 
 14 
 
 29 
 
 15 
 
 14 
 
 17 
 
 30 
 
 66 
 
 OFF./ C *! P ' G MACHINE 
 
 00 
 
 14 
 
 14 
 
 30 
 
 15 
 
 14 
 
 17 
 
 20 
 
 67 
 
 SERV. 1ND. MACHINES 
 
 00 
 
 14 
 
 14 
 
 29 
 
 15 
 
 14 
 
 17 
 
 30 
 
 68 
 
 ELEC. TRANS APPARAT. 
 
 00 
 
 14 
 
 14 
 
 29 
 
 15 
 
 14 
 
 17 
 
 30 
 
 69 
 
 HOUSEHOLD APPLIANCES 
 
 00 
 
 14 
 
 14 
 
 29 
 
 . 15 
 
 14 
 
 17 
 
 30 
 
 70 
 
 ELEC. LIGHT • C . . .EQUIP. 
 
 13 
 
 14 
 
 14 
 
 29 
 
 15 
 
 14 
 
 17 
 
 30 
 
 71 
 
 RADIO/ TV/ COM. E3UIP. 
 
 00 
 
 14 
 
 14 
 
 29 
 
 15 
 
 14 
 
 17 
 
 30 
 
 72 
 
 ELEC. COMPONENTS. .. 
 
 00 
 
 14 
 
 14 
 
 19 
 
 15 
 
 14 
 
 17 
 
 20 
 
 73 
 
 H1SC. ELEC. ..SUP-LIES 
 
 00 
 
 14 
 
 14 
 
 29 
 
 15 
 
 14 
 
 17 
 
 30 
 
 7k 
 
 MOTOR VEHICHLES CQ'T 
 
 13 
 
 14 
 
 14 
 
 19 
 
 15 
 
 14 
 
 17 
 
 20 
 
 75 
 
 AIRCRAFT AND » A fi T S 
 
 00 
 
 14 
 
 14 
 
 19 
 
 15 
 
 14 
 
 17 
 
 20 
 
 76 
 
 OTHER TRANS. EQUIPMENT 
 
 00 
 
 14 
 
 14 
 
 19 
 
 15 
 
 14 
 
 17 
 
 20 
 
 77 
 
 PROFESSIONAL.'. .SUP PL IE 
 
 00 
 
 14 
 
 14 
 
 29 
 
 15 
 
 14 
 
 17 
 
 30 
 
 78 
 
 0P1CAL EQUIP. AND 
 
 00 
 
 14 
 
 14 
 
 19 
 
 15 
 
 14 
 
 17 
 
 20 
 
 79 
 
 MISC. MANUF ACTUPINu 
 
 00 
 
 14 
 
 14 
 
 29 
 
 15 
 
 14 
 
 17 
 
 30 
 
 80 
 
 RAILROADS AND ... SERV ' S 
 
 00 
 
 13 
 
 13 
 
 30 
 
 15 
 
 22 
 
 00 
 
 18 
 
 81 
 
 ...HIGHWAY PASS. T R A N . 
 
 00 
 
 13 
 
 13 
 
 30 
 
 15 
 
 22 
 
 00 
 
 1b 
 
 82 
 
 MOTOR FREIGHT TRANS 
 
 00 
 
 13 
 
 13 
 
 30 
 
 15 
 
 22 
 
 00 
 
 18 
 
 83 
 
 WATER TRANSPORTATION 
 
 00 
 
 13 
 
 13 
 
 30 
 
 15 
 
 22 
 
 00 
 
 18 
 
 81* 
 
 AIR TRANSPORTATION 
 
 00 
 
 13 
 
 13 
 
 30 
 
 15 
 
 21 
 
 00 
 
 18 
 
 85 
 
 PIPE LINE TOANSPORTA'N 
 
 00 
 
 22 
 
 22 
 
 00 
 
 15 
 
 22 
 
 00 
 
 18 
 
 86 
 
 TRANSP0RTA10N SERVICES 
 
 00 
 
 90 
 
 00 
 
 30 
 
 15 
 
 22 
 
 00 
 
 18 
 
 87 
 
 COHVNS EXCEPT RADIO... 
 
 00 
 
 ?7 
 
 27 
 
 30 
 
 14 
 
 22 
 
 14 
 
 23 
 
 88 
 
 RADIO AND TV BROADCAST 
 
 00 
 
 27 
 
 27 
 
 00 
 
 14 
 
 22 
 
 14 
 
 23 
 
 89 
 
 WATER AND SAN. SERV'S 
 
 00 
 
 27 
 
 27 
 
 30 
 
 14 
 
 22 
 
 14 
 
 23 
 
 90 
 
 -WHOLESALE AND RETAIL.. 
 
 00 
 
 27 
 
 27 
 
 26 
 
 26 
 
 22 
 
 14 
 
 23 
 
 91 
 
 FINANCE AND INSURANCE 
 
 00 
 
 27 
 
 27 
 
 30 
 
 14 
 
 22 
 
 14 
 
 23 
 
 92 
 
 REAL ESTATE AND RENTAL 
 
 00 
 
 27 
 
 27 
 
 30 
 
 14 
 
 22 
 
 14 
 
 23 
 
 93 
 
 HOTELS AND EXCEPT... 
 
 00 
 
 27 
 
 27 
 
 26 
 
 52 
 
 52 
 
 14 
 
 23 
 
 9^ 
 
 BUSINESS SERVICES 
 
 00 
 
 27 
 
 27 
 
 30 
 
 14 
 
 22 
 
 14 
 
 23 
 
 95 
 
 AUTO. REPAIR AND SERV. 
 
 00 
 
 27 
 
 27 
 
 30 
 
 14 
 
 22 
 
 14 
 
 23 
 
 96 
 
 AMUSEMENTS 
 
 00 
 
 27 
 
 27 
 
 26 
 
 26 
 
 22 
 
 14 
 
 23 
 
 97 
 
 MED./ ED. SERV'S AND.. 
 
 00 
 
 27 
 
 27 
 
 ?6 
 
 26 
 
 22 
 
 14 
 
 23 
 
 98 
 
 FED. GOV'T ENTERPRISES 
 
 00 
 
 27 
 
 27 
 
 30 
 
 14 
 
 22 
 
 14 
 
 23 
 
 99 
 
 STATE AND LOCAL EN'S 
 
 00 
 
 27 
 
 27 
 
 30 
 
 14 
 
 22 
 
 14 
 
 23 
 
 100 
 
 BUS. TRAV AND GIFTS 
 
 00 
 
 00 
 
 00 
 
 00 
 
 00 
 
 00 
 
 00 
 
 00 
 
 101 
 
 OFFICE SUPPLIES 
 
 00 
 
 00 
 
 00 
 
 30 
 
 00 
 
 00 
 
 00 
 
 00 
 
 102 
 
 PERS. CONSUMP'N EXPEN. 
 
 00 
 
 28 
 
 28 
 
 28 
 
 28 
 
 28 
 
 28 
 
 28 
 
 103 
 
 bROSS...CAP. FORMATION 
 
 00 
 
 00 
 
 00 
 
 30 
 
 00 
 
 00 
 
 00 
 
 00 
 
 10U 
 
 NET INVENTORY CHANGE 
 
 00 
 
 00 
 
 00 
 
 00 
 
 00 
 
 00 
 
 00 
 
 00 
 
 105 
 106 
 
 NET EXPORTS 
 
 00 
 
 00 
 
 00 
 
 00 
 
 00 
 
 00 
 
 00 
 
 00 
 
 FED. GOV'T DEFENSE 
 
 00 
 
 13 
 
 13 
 
 26 
 
 26 
 
 22 
 
 14 
 
 23 
 
 107 
 108 
 
 FED. GOV'T OTHER 
 
 00 
 
 27 
 
 27 
 
 30 
 
 14 
 
 22 
 
 14 
 
 23 
 
 STATE.. .GOV'T. . .EDUC'N 
 
 00 
 
 27 
 
 27 
 
 26 
 
 26 
 
 22 
 
 14 
 
 23 
 
 109 
 
 STATE HEALTH/ SAN. 
 
 00 
 
 27 
 
 27 
 
 26 
 
 26 
 
 22 
 
 14 
 
 23 
 
 110 
 
 STATE GOV'T SAFETY 
 
 00 
 
 27 
 
 27 
 
 00 
 
 14 
 
 22 
 
 14 
 
 23 
 
 111 
 
 STATE. ..GOV'T. . .OTHER 
 
 00 
 
 27 
 
 27 
 
 30 
 
 14 
 
 22 
 
 14 
 
 23 
 
 56 
 
K 
 O 
 
 Eh 
 
 u 
 w 
 
 CQ 
 
 rH 
 
 O 
 
 co 
 o 
 
 O 
 
 H 
 CQ 
 
 >H 
 1-3 
 
 CQ 
 
 a 
 o 
 
 CO 
 
 s 
 o 
 
 H 
 E-t 
 C_> 
 < 
 CO 
 
 g 
 
 CO 
 
 § 
 o 
 
 w 
 o 
 
 o 
 
 Eh 
 
 H 
 
 o 
 
 o 
 
 O 
 O 
 
 o 
 o 
 
 o 
 
 o 
 
 o 
 o 
 
 on 
 
 H 
 
 on 
 
 H 
 
 rH 
 
 H 
 
 o 
 
 o 
 
 o 
 o 
 
 o 
 
 o 
 
 o 
 
 o 
 
 on 
 H 
 
 on 
 
 H 
 
 
 on 
 
 H 
 
 -3" 
 H 
 
 m 
 
 o 
 o 
 
 o 
 o 
 
 00 
 H 
 
 on 
 
 on 
 H 
 
 en 
 H 
 
 on 
 
 H 
 
 on 
 
 H 
 
 O 
 
 o 
 
 o 
 o 
 
 o 
 o 
 
 on 
 
 H 
 
 CO 
 H 
 
 on 
 H 
 
 on 
 
 H 
 
 on 
 
 H 
 
 1 OJ 
 H 
 
 en 
 
 H 
 
 o 
 o 
 
 o 
 o 
 
 on 
 
 H 
 
 on 
 
 H 
 
 on 
 
 H 
 
 on 
 
 H 
 
 on 
 
 rH 
 
 i a 
 
 O 
 O 
 
 o 
 o 
 
 o 
 o 
 
 on 
 
 H 
 
 o 
 o 
 
 o 
 o 
 
 o 
 o 
 
 o 
 
 o 
 
 o 
 
 H 
 
 on 
 
 H 
 
 o 
 o 
 
 o 
 o 
 
 on 
 
 H 
 
 en 
 
 H 
 
 o 
 o 
 
 o 
 o 
 
 o 
 o 
 
 1 On 
 
 on 
 
 H 
 
 o 
 o 
 
 o 
 o 
 
 o 
 o 
 
 o 
 o 
 
 o 
 o 
 
 o 
 
 o 
 
 o 
 o 
 
 CO 
 
 o 
 o 
 
 o 
 o 
 
 o 
 o 
 
 o 
 o 
 
 o 
 
 o 
 
 o 
 o 
 
 o 
 o 
 
 H 
 
 o 
 
 t— 
 
 o 
 
 o 
 
 o 
 o 
 
 o 
 o 
 
 o 
 o 
 
 o 
 o 
 
 o 
 o 
 
 H 
 
 o 
 
 o 
 o 
 
 VO 
 
 H 
 O 
 
 o 
 o 
 
 o 
 o 
 
 H 
 
 o 
 
 H 
 
 o 
 
 H 
 
 o 
 
 o 
 
 o 
 
 o 
 o 
 
 ir\ 
 
 H 
 
 H 
 
 H 
 
 o 
 
 o 
 
 o 
 
 H 
 H 
 
 H 
 
 o 
 
 H 
 H 
 
 H 
 H 
 
 H 
 
 H 
 
 I -d- 
 
 H 
 O 
 
 H 
 
 o 
 
 o 
 o 
 
 H 
 
 o 
 
 o 
 
 H 
 O 
 
 H 
 O 
 
 H 
 
 O 
 
 on 
 
 O 
 H 
 
 o 
 o 
 
 o 
 o 
 
 o 
 o 
 
 o 
 
 H 
 
 O 
 O 
 
 O 
 O 
 
 o 
 o 
 
 CVJ 
 
 H 
 H 
 
 OJ 
 
 o 
 
 o 
 
 o 
 
 CM 
 
 o 
 
 OJ 
 O 
 
 OJ 
 O 
 
 OJ 
 O 
 
 OJ 
 
 o 
 
 rH 
 
 O 
 
 o 
 o 
 
 o 
 o 
 
 OJ 
 
 o 
 
 OJ 
 
 o 
 
 OJ 
 O 
 
 OJ 
 
 o 
 
 OJ 
 
 o 
 
 
 H 
 
 CVJ 
 
 on 
 
 ,=r 
 
 ir\ 
 
 vo 
 
 c— 
 
 CO 
 
 u 
 o 
 
 -p 
 
 o 
 (U 
 
 w 
 
 rH 
 O 
 
 Cm 
 
 on 
 
 i 
 
 OJ 
 
 Eh 
 
 O 
 
 -P 
 
 u 
 
 «m 
 0) 
 
 S7 
 
A. 3 DISPERSION FACTORS FOR SMALL-MAGNITUDE FIGURES 
 
 Often the uncertainty on a column of figures, X- would be described 
 in terms of "a dispersion factor D. for x. = B increasing to a dis- 
 persion factor D for the smallest value reported." Taking this lower 
 bound to be X- = A where A = $10 for the 196? U.S. input-output 
 tables, and assuming a linear dependence of D(x) on log(x) we obtain 
 the following expression for D as a function of x-« Let D(x) = a log(x) 
 + b where a = (D g - D )/(log A - log 3) and b = (D 1 log A - D g log B)/ 
 (log A - log B). It is easy to verify that D(x) takes values D x and Z 2 
 at x = B and x = A respectively. 
 
 Obviously this is a crude approximation, but it actually may be 
 even too refined when viewed from the perspective of the person estima- 
 ting the uncertainty. 
 
 58 
 
A.U PROBABLE VALUES OF "ZERO" ELEMENTS 
 
 Since few transactions can be defined to be zero, the published 
 figures truncated at $10 dollars may be misleading. It is probably- 
 true that if we examined in detail the transactions of all firms in the 
 U.S. defined by a particular transaction cell in the 1-0 table, we would 
 find at least one nonzero transaction. Therefore the following approxi- 
 mation was used to estimate the probable distribution of nonzero values 
 between the lower and upper bounds [0, $10 ]. 
 
 Let X be the absolute value of a normal random variable Y with 
 mean and a = 10/ . Then X takes nearly all its values between 
 and 10 . By truncating X at 10 , in the sense that larger values are 
 discarded and resampled, the resulting random variable takes all of 
 its values in [0, 10 ] with the great bulk of its unit probability ac- 
 cumulated near zero. 
 
 For direct energy transactions, the cutoff was 10 ^Btu, which 
 corresponds to approximately the same dollar value. 
 
 Details of the "folded normal" distribution are given in Appendix C. 
 
 59 
 
Appendix B. Sector Definitions 
 
 TABLE B-l. 30 Sector Model 
 
 30 SECTOR 
 
 MODEL BEA SECTOR 
 
 1. 7 Coal 
 
 2. 8 Crude Oil and Natural Gas 
 
 3. 31.01 Refined Petroleum Products 
 
 k. 68.01 Electric Utilities 
 
 5- 68.02 Natural Gas Utilities 
 
 6. 1-k Agriculture 
 
 7. 5-10 Mining 
 
 8. 11-12 Construction 
 
 9. Ik, 15, 29 Food and Drugs 
 
 10. 16-19 Textiles and Apparel 
 
 11. 20-26 Wood and Paper Products 
 
 12. 27, 28, 30-32 Paint, Plastics and Oil Products 
 
 13. 33, 3^+ Leather and Shoes 
 
 lk. 35, 36 Stone, Clay and Glass Products 
 
 15. 37-^2 Metals and Metal Products 
 
 16. 1+3-52 Machinery 
 
 17. 53-58 Electrical Equipment and Appliances 
 
 18. 59-61 Cars, Planes and Transport Equipment 
 
 19. 62-6U, 13 Miscellaneous Manufacturing 
 
 20. 65.OI Rail Transport 
 
 21. 65.02 Local Passenger Transport 
 
 22. 65.03 Truck Transport and Warehousing 
 
 23. 65. 0k Water Transport 
 
 2k. 65.05 Air Transport 
 
 25. 65.06 Pipeline Transport 
 
 26. 66-67 Radio, TV, Communications 
 
 27. 69 Wholesale and Retail Trade 
 
 28. 70-71 Finance, Insurance and Real Estate 
 
 29. 65.07, 68.03, 72-79 • • • Services 
 
 30. 81-82 Business Travel and Office Supplies 
 
 60 
 
TJ *t? 0) 
 
 § S3 
 
 v< W) C 
 
 ,7 
 
 I. "1 
 
 <J 
 
 O 
 
 
 W 01 
 
 c 
 
 SS 
 
 •3 
 
 U) H* ♦> 
 
 "J 
 
 oi i, n 
 
 H>> ^ 
 
 u) tl b 
 
 •rH 
 
 
 •H H 
 
 "3 rH 
 
 
 >-. *J C 
 
 G 0) 
 
 > 
 
 P. C -r. 
 v« oi 4 
 
 01 *> 
 
 >-. 
 
 B-T3. 
 
 01 
 
 «-• p >- 
 
 t : u a,' 
 
 VI 
 
 oi d 
 
 
 o 
 
 -1 
 
 01 OJ 44 
 
 c 
 
 c *» o 0> 
 
 O ft O > -rH 
 
 3 Oi O d H 
 
 TJ > rH I- P, 
 
 ■3 3 
 
 to TJ 
 
 >•£ p 
 
 •H O *H 4J In 
 
 2 s < o a. 
 
 p. 3 
 
 -H I* 
 
 45 
 
 0> CJ Lt o> c ci 
 to -ri Oi 4* *rt -H 
 
 fl ^ o] w t* 
 Of 01 *> 3 Vh 
 
 <:hnno 
 
 v II tl II 
 
 ♦>♦»+>♦> 
 el d a) ol 
 
 «■ f f <> 
 
 M 10 0) M 
 
 r-i cni on.-*; unno t— m oooooooooo 
 
 NI , 1Jinv0f-0)mOH0|l , >Jlf>0t-l0(?iOHNnjOOOOO0O>flt-O0vOrtfgl , >m\0l-(0l>H(\IV0^0)arttll^MOW 
 
 j3j3^*^^s^s^if\ir\\r\tf\if\\f\u\tr\if\ir\\o\o\o*&*£> M}^ -vot--r— r-r— c— r--t--t--t— coco 
 
 un u\v\ ir\ m in irv co nosononOc 1 — t— cocococo 
 
 NO VO SO NO NO NO NO VO OS ON C\ 0\ O ON ON ON ON Ov 
 
 
 
 cm 
 
 o 
 •p 
 
 CJ 
 <D 
 CO 
 
 H 
 O 
 
 rH 
 
 M I M C_> 
 
 I I I I I I I IUUOUHH 
 
 OUMMMI-IMMUMl-l 
 
 +J T3 
 
 
 ♦> 
 
 
 
 •H 
 
 H-> 
 
 r-i 0) 
 
 
 
 
 ■0 
 
 
 <J 
 
 tH 
 
 ■rH V. 
 
 
 c 
 
 5 
 
 
 ■rH 
 
 H 
 
 +2 
 
 
 
 
 rH 
 
 •rH 
 
 3^~ 
 
 
 -a 
 
 
 
 
 H-> 
 
 c 
 
 
 § 
 
 to 
 
 
 P 
 
 3 
 
 O O 
 
 
 c 
 
 
 3 
 
 
 ■r* tH 
 
 
 
 •rH 
 
 
 
 CI 
 
 rH +-> CO 
 
 <j o a 
 
 
 
 c 
 
 
 O 
 
 ■rH 
 
 
 oi "3 
 
 
 
 
 rH 
 
 CJ 3 o 
 
 
 CH 
 
 cl» 
 
 Ih 
 
 H-> 
 
 01 TJ O 
 
 
 „"- 1 o 
 
 CI 
 
 u 
 
 +> 
 
 CI 
 
 •H 01 4-> 
 
 
 sou 
 
 |H 
 
 
 O 
 
 01 
 
 0; -- A 
 
 
 a t. 
 
 
 HJ 
 
 0> 
 
 rH 
 
 
 •rH +) rj 
 
 i 
 
 
 
 01 
 
 0) 01 0> 
 
 
 C 01 1) 
 
 H 
 
 01 
 
 
 rH Ih 01 
 
 
 Ti °<V 
 
 e 
 
 
 
 In 
 
 rO O ^ 
 
 
 a i. 
 
 h 
 
 h-> 
 
 r-\ 
 
 CD 
 
 a) ^ 
 
 
 01 0i 
 
 
 
 3 
 
 •r-l 
 
 01 
 
 * h 
 
 
 h -a > 
 
 4 3 a 
 
 t, 
 
 
 CO 
 
 rH 
 
 01 01 u 
 
 
 
 H 
 
 I*: 
 
 CI 
 
 r: .* j.: 
 
 
 o i. o 
 o o o 
 
 01 
 
 a. 
 
 d 
 
 o 
 
 3 
 
 oi o -ty 
 cc o o 
 
 i 
 
 
 
 'J 
 
 
 rH 
 
 CM 
 
 M 
 
 H 
 
 rH 
 
 t>-00 1 
 
 O 
 
 O 
 
 o 
 
 O 
 
 O 1 1 
 
 
 M a") CO CO 00 
 
 
 
 i^i NO NO NO vO 
 
 
 
 
 
 
 > 
 
 
 to 
 
 
 
 
 
 
 rH 
 
 
 c 
 
 
 
 
 
 
 01 
 
 
 •H 
 
 
 
 
 
 
 to 
 
 
 c 
 
 
 
 
 
 
 r? 
 
 
 
 
 
 
 
 ■ 
 
 H-> 
 
 a. 
 x: to 
 
 
 &3 
 
 
 
 
 
 o 
 
 Ul c 
 
 
 t. rH 
 
 
 
 
 
 3 
 
 « -H -H 
 
 
 0] 01 
 
 
 
 
 
 -a 
 
 hJJ V-i C 
 
 bfl 
 
 3 CJ 
 
 
 
 
 
 e 
 
 WO -H 
 
 B 
 
 •H 
 
 eg 
 
 
 
 
 
 a 
 
 3 o a) co 
 
 C 
 
 13 
 
 3 oi 
 
 
 
 
 
 .* 
 
 TJ Ih 0) 
 
 B 
 
 
 
 
 
 CI 
 
 s *&§ 
 
 
 to 
 
 
 
 
 
 o 
 
 09 
 
 60 -H 
 
 
 
 
 
 ■*» 
 
 rH CD >, 
 
 H( 
 
 C r-l 
 
 
 
 
 
 n 
 
 t. 
 
 
 
 
 
 
 n 
 
 rH o> aj o 
 
 
 
 .St 
 
 
 
 
 
 > 
 
 a! s: u rH 
 
 
 
 
 0C 
 C 
 
 
 
 S 5 fl 3 
 
 rH 
 
 01 
 
 U oi 
 
 Vh 
 
 
 
 
 -H 
 
 
 ()C, o 
 
 ■p 
 
 >, 
 
 
 
 c 
 
 0) 
 
 •a 
 
 .-( - Vh 
 
 J OH r. 
 
 01 
 
 <J T) 
 
 
 
 o 
 
 ^ 
 
 c 
 
 
 rH U 
 
 
 
 
 o 
 
 a) 
 
 O C cd 01 
 
 •H 3 Ch Cm 
 
 
 ci cd 
 
 .-> 
 
 4J 
 
 ♦J 
 
 a. 
 
 
 3 
 
 
 ttj 
 
 0) 
 
 
 
 >; 
 
 1h 3 
 
 XJ CO 
 
 01 
 
 0) 
 
 a 
 
 o 
 
 CJ 
 
 S h** 'S 
 
 o 
 
 S3 
 
 o 
 
 .C 
 
 j3. 
 
 
 
 •H 
 
 rH 
 
 o 
 
 H-> 
 
 c- 
 
 rH 
 
 r, 
 
 cu 
 
 u 
 
 +> 
 
 CO 
 
 rl U) (J 
 
 CJ 
 
 01 *<H 
 
 01 
 
 o 
 
 
 o 
 
 0' 
 
 oi cu — * n 
 
 Ch 
 
 n ri 
 
 
 d 
 
 >H 
 
 1) 
 
 > 
 
 .r; in i. u 
 
 
 o c; 
 
 al 
 5 
 
 P. 
 CQ 
 
 -3 
 
 bl 
 
 k5 
 
 +-> o iifl in 
 5 Cm vjTh 
 
 ;- 
 
 
 e a 
 
 •h J-. TJ v« ty 
 
 i-h aj o o to ,o r-i 
 
 O, CO V- 4) CJ -rt 
 
 Cl to P» Ji ^-» -P 
 
 0) jJ -p CO CO 
 
 X (j (J w 
 
 q t; cj o t? 
 
 ^-« 13 to -H CJ ■--* 
 
 <u T) J-r cu 
 
 
 01 
 
 
 r^ 
 
 CO 
 
 a. 
 
 w 
 
 
 i; 
 
 (0 
 
 •H 
 
 ■3 
 
 *J 
 
 o 
 
 '31 
 
 ai 
 
 o 
 
 -1 
 
 1) '3 O. -3 O 
 
 i3 p, 10 3 J 
 
 o 
 
 cu o 5 *> 
 
 
 «H 
 
 11 
 
 0) 
 
 cl 
 
 r] Oj lH 
 
 TJ In In W 
 
 a 3 o d 
 
 
 
 <> 
 
 
 Tl 
 
 
 3 
 
 O 
 
 -d 
 
 s 
 
 •rH c c o 
 
 
 t) 
 
 
 p 
 
 J4 d 0) 
 
 
 il 
 
 •u 
 
 c 
 
 Q) 
 
 and 
 co m 
 and 
 llan 
 
 
 a 
 
 3 
 
 o 
 o 
 
 f) 
 
 rH 
 
 rH 
 
 
 
 Sfl 
 
 11 
 
 
 rH 
 
 c 
 
 U r-H 0> 
 
 rH 
 
 0> 
 
 0) 
 
 01 
 
 
 ■o a a o 
 
 cfl 
 
 CJ 
 
 rO 
 
 rd 
 
 -(I 
 
 O rO W 
 
 n. 
 
 01 
 
 i 
 
 o 
 
 1, 
 
 O O rH •H 
 
 p. 
 
 
 o 
 
 o 
 
 •j: 
 
 
 , i 
 
 s 
 
 '3 
 
 d 
 
 ^J 'U 
 
 it 
 
 o 
 
 
 UJ 
 
 
 
 
 
 f0 
 
 
 
 p. 
 
 10 
 
 1." f H 
 
 i> 
 
 ■-( 
 
 
 
 
 
 Im 
 
 
 c; 
 
 ■c 1 c: 
 
 ) 
 
 a a 
 
 1. 'O 
 
 
 vC 
 
 rfl 
 
 S 01 
 
 5 
 
 u 
 
 T) 
 
 rH 
 
 CO 
 
 
 r 1 Si 
 
 "1 
 
 a 
 
 d 
 n 
 
 5° ^ 
 
 CM 
 
 fj flj 
 
 n' 
 
 
 rO 
 
 ■«^ cj 
 
 
 * CO 
 
 i«i 
 
 t. 
 
 rn 
 
 P H 
 
 ♦* 
 
 CO -P 
 
 
 OJ 
 
 01 
 
 G 1 
 
 CO 
 
 i.fl ^ 
 
 
 ,-< r. u u o 
 
 ■rd'OrHd4)0.rH 
 0) ^3 p. 
 
 o tc ■*-> CO 
 
 hj r i; a h h n * 
 
 
 
 oi a 
 
 
 
 a p 
 
 
 
 rH S 
 
 
 
 d 
 
 
 
 r< «0 
 
 
 
 3 
 
 
 Ml 
 
 +» • 
 
 
 c 
 
 o o 
 
 M 
 
 rH 
 
 a « 
 
 C 
 
 3 
 
 p 
 
 
 *J 
 
 10 - 
 
 1h 
 
 o 
 
 CO 
 
 3 
 
 U 
 
 TJ +J 
 
 H-> 
 
 Ch 
 
 OJ 3 
 
 Cl 
 
 3 
 
 •*-> C 
 
 (f 
 
 c: 
 
 d 
 
 l-H 
 
 3 
 
 o « 
 
 3 
 
 9 
 
 •H CO 
 
 t: 
 
 
 r. *» 
 
 II 
 
 
 ^1 rH 
 
 
 d o 
 
 
 01 
 
 *H r3 
 
 r-1 
 
 HJ 
 
 
 01 
 
 u 
 
 -a • 
 
 01 
 
 H-> 
 
 y 
 
 5r» 
 
 
 U J 
 
 » 2 
 
 r, -r. °< 
 
 Tl 
 
 o 
 
 g 
 
 c, 
 
 u 
 
 Oi ^ 
 
 s s 
 
 M O O C oJ <H *H 
 
 -O TJ 
 
 »h t: c 
 
 
 u 
 
 (1 G 
 
 o 
 
 
 cd 
 
 a d 
 
 i?i:- 
 
 
 3 
 
 CO 01 
 
 ■o i: 
 
 1 fit 
 
 ■H 
 
 01 
 
 3. 
 
 x o ii, ci; p', c~ i\, ii ill c','; .5 fj; .% im ci t7j rv, p. x .-a co 
 
 .3 d - 
 
 J ' , C!J 
 
 '« » 
 
 I rH CN| CN J 
 
 I.N ' O t JJ U* 
 
 Mrr, 
 
 OJ 
 
 r^ 
 
 3 
 
 H evi I r»N t/N 
 
 I I I I I I I NO [— 
 
 to on o rH <\j cu un\o t— nocin rnrvifi-t iano t-cno\o hcj 'o.j ia\o f-* o ; o.-ii\i m.jij 
 
 ' *NJ C\J <NJ CNJ rvj C\l CNJ cNI C\J c\l *^ »^ r ^ <*^ '^ ( ^ c*i r-i ro f*l ^-f -J . h* -* -^* -hT 
 
 r-C rH rH rH rH r~\ 
 
Appendix C. Treatment of Zero Values 
 
 In section 1 the variance of a "folded" normal is computed while 
 section 2 details the relationship between the variance of a lognormal 
 and its dispersion factor. 
 
 C.l Variance of a "Folded" Normal Distribution 
 
 The variance of any random variable, X, is given by Var(X) = 
 E(X 2 ) - (E(X)) 2 , where E denotes expected value. If Y is N(0,l) then 
 1 = Var (Y) = E(Y 2 ) - 2 so E(Y 2 ) =1. Z is said to be "folded" N(0,l), 
 if Z * ABS(Y). Then E(Z 2 ) = E(Y 2 ) = 1. Therefore Var (Z) = E(Z 2 ) - (E(z)) 2 . 
 
 But 
 
 2 2 
 
 -t -t I °° 
 
 E(Z) . -i- te" 2 " d = -§rl- ? u-2- 
 
 ; r 2n . 2ii j ; o 2n 
 
 therefore Var (Z) = 1 - (E(z) ) 2 = 1 - -| 
 
 If we fold a normal random variable with three sigmas equal to b then the 
 variance computed above is multiplied by a factor of (— ) . Conversely 
 
 if we know the variance V of a folded normal then one sigma of the underlying 
 normal is \ V 
 
 -I 
 
 C.2 Variance of a Lognormal Distribution 
 
 If a given cell is lognormal with published mean M and three sigma 
 
 2 
 
 dispersion factor D, then we sample for this cell by exponentiating a N(a,3 ) 
 
 2 
 . ,., , „ InD , , „. In D 
 random variable where 8 = -rr— and a = InM - „ . 
 
 62 
 
The variance of a lognormal random variable with parameters a and 3 is 
 
 given by V = e (e -l) = M (e -l). Substituting — r— for $ we obtain 
 
 2 
 
 V = M * exp — - — -1 . Conversely, we can solve this equation for D: 
 
 D = exp (3* lnd+V/M 2 )' 
 
 Incidentally, at the changeover point from normal to lognormal we have 
 
 ' " \ 
 ABS -rr/ r .h or —? = .0178. At this point D = I.U89. As we cross the changeover 
 
 point from normal to lognormal we switch from a normal with range between 
 60% and ll+0% of the published mean value to a lognormal with D = . IU89 and 
 1/D = .67. 
 
 63 
 
APPENDIX D. RANDOM NUMBER GENERATOR VERIFICATION 
 By Robert Bohrer and Dan Putnam 
 
 D.l INTRODUCTION 
 
 The present Monte Carlo study of the sensitivity of Input-Output data 
 to stochastic estimation errors requires the use of a fast, reliable nor- 
 mal random number generator. In the present case each Monte Carlo trial 
 requires roughly ten thousand random numbers to perturb the parameters 
 in the Input-Output model. Since several hundred trials are required to 
 obtain useful results, speed is an important consideration in the choice 
 of a generator for this application. Furthermore, a necessary condition 
 for the validity of the results is, of course, that the random inputs 
 conform to the modeling assumptions. I n this case the required inputs 
 are independent, normally distributed random numbers. 
 
 The generators in the International Mathematical and Statistical Li- 
 lt 
 braries were given special consideration for use in the Monte Carlo study 
 
 because of the good reputation of IMSL and the availability of IMSL at 
 most large IBM installations. One generator, GGNRF, was especially ap- 
 pealing since it was designed specifically as a fast normal random num- 
 ber generator and had already been optimized and coded in assembler 
 language for efficiency. The one flaw in the qualifications of GGNRF 
 was that the existing documentation of its statistical properties was 
 inadequate for our needs. Although the distributional properties had 
 been checked with several goodness of fit tests, independence had been 
 
 Available from International Mathematical and Statistical Libraries 
 (IMSL) Inc., Sixth Floor, GNB Building, 7500 Bellaire Boulevard, Houston, 
 Texas 77036. 
 
checked only up to five lags. Documentation of these tests may be found 
 in Kuki (197*0. While it is impossible to test all aspects of randomness, 
 it is prudent to examine at least those aspects most important to the 
 particular application. The present Monte Carlo study requires inde- 
 pendence among the inputs to each Monte Carlo trial and among the 
 separate trials as well. The tests described in this paper were designed 
 by the first author to examine both types of independence. Tests of nor- 
 mality were also included for the sake of completeness. 
 
 These properties may be tested by selecting several seeds for the 
 generator and examining three sequences obtained from each. The first 
 two sequences are chosen so that in the actual simulation they would oc- 
 cupy corresponding positions in the inputs to consecutive simulation 
 runs. Thus, the first N numbers drawn from a seed constitute the first 
 sequence. Then, as many more numbers are generated as would be needed 
 to complete one simulation run. The second sequence then consists of 
 the next N numbers drawn. To examine the independence between these 
 first two sequences, a third sequence of 2N numbers is formed by shuff- 
 ling the other two sequences so that the odd numbered entries are taken 
 in order from the first sequence and the even numbered entries are taken 
 in order from the second sequence. 
 
 It is desirable to use tests which are sensitive to as many depar- 
 tures from the hypotheses as possible. The methods used here allow 
 tests of the independence .of X(t) and X(t+L) for all lags L less than 
 
 * 
 
 For a discussion of random number generation and testing see Jansson (1966) 
 
 65 
 
or equal to the sample size N. For this reason and because the statis- 
 tical power of these tests increases with sample size, the value of N 
 used in defining the sequences above should be as large as possible. 
 While it would be desirable to make N as large as the number of inputs 
 to a Monte Carlo trial, this possibility was precluded by considerations 
 of the availability of computing resources in the statistical analyses 
 of the random number samples. In this study N = 102U was chosen with 
 these considerations in mind and because of the efficiency of working 
 with a power of 2 in Fast Fourier Transformation. Five seeds were 
 selected for the tests that follow, again with this number being de- 
 termined partly by considerations of available computing resources 
 versus the quantity of information generated. 
 
 Section 2 details the tests performed on the shuffled sequences 
 obtained from the five seeds to check independence between Monte Carlo 
 trials. Section 3 describes the tests on the two unshuffled sequences 
 from each seed to examine the independence of the inputs to a single 
 trial. Finally, section h documents the test performed on the un- 
 shuffled sequences to check that the samples are normally distributed 
 with mean zero and variance one. 
 
 66 
 
D.2 INDEPENDENCE BETWEEN MONTE CARLO SAMPLES 
 
 To test the independence of consecutive runs from the Monte Carlo 
 experiment, the shuffled sequences from the five seeds were examined. 
 Given the hypothesis of independence between simulation runs, the sample 
 autocovariances at odd lags, L, of the shuffled sequences should be 
 independent and standard normal (i.e., N(o,l)) when multiplied by 
 
 1/20U8-L. A test of the independence of consecutive Monte Carlo trials 
 
 may then be made by using the Kolmogrov-Srairnov (K-S) statistic to 
 
 compare the sample distribution function of the adjusted autocovariances 
 
 at odd lags to the standard normal distribution function. The sample 
 
 used for this test was therefore s 1 *** s 2 56» S i = AC 2i-l * v/20U8-(2i-l) 
 
 where the AC~. , are the odd lag autocovariances. Table 1 lists the five 
 2i-l 
 
 K-S statistics and P-levels obtained from the five shuffled sequences 
 
 in this way. The tabled statistics represent the square root of the sample 
 
 That the mean is and the variance is 1/20U8-L follows from simple 
 calculation with expectations. The asymptotic normality and independence 
 follow from careful application of a multivariate central limit theorem 
 such as theorem 9.2.3 in Wilks (1962). For example, normality of the 
 lag L covariance estimate is shown by applying the theorem to the two 
 sequences 
 
 (X 1*W ••• X L* X 2L' X 2L + 1* X 3L + 1> •" ) 
 
 and (Xl+^sl+I* •" X 2L* X 3L' X 3L+1* X UL+1 ' •** ) ' 
 
 If the covariance estimates are adjusted for the sample mean, then the 
 tests for the hypothesized means and covariances can be made separately, 
 each having valid significance level, even under certain natural dis- 
 crepancies from the other hypothesis. Also, a theorem in section 20.6 
 of Cramer (19U6) shows that the large sample distribution theory is 
 exactly the same as described above. 
 
 67 
 
size times the maximum absolute difference between sample and hypothesized 
 distribution functions. The P-level is defined as the probability that a 
 truly normal sample would have achieved a larger value for the K-S statis- 
 tic than the value actually observed. Given the independence hypothesis, 
 the P-levels should be independent and uniformly distributed on the unit 
 interval. This fact allows the use of the following summary statistic: 
 
 if P..«'«P are independent and uniform on the unit interval, then 
 
 n * 
 
 -2*£ ln(P.) is chi square with 2n degrees of freedom. This sample 
 
 1 X 
 statistic and its own P-level are included in Table 1 along with the 
 
 K-S statistics and their P-levels as an additional check and summary. 
 
 TABLE 1 
 
 
 Statistic 
 
 P-level 
 
 1 
 
 .Ihk 
 
 .6U 
 
 2 
 
 .869 
 
 M 
 
 3 
 
 .5^8 
 
 .92 
 
 k 
 
 .807 
 
 .53 
 
 5 
 
 .772 
 
 .59 
 
 -2*Z ln(P ) 
 
 5.03 
 
 .89 
 
 This derives from the fact that if P is uniform on [0,1], then -2*ln(P) 
 is exponentially distributed with distribution function l-e'X'2^ 
 The result now follows by verifying that the associated density function 
 is that of a chi-square random variable with two degrees of freedom. 
 
 68 
 
A plot of the sample distribution function with the worst fit (from 
 seed #2 in this case) is shown in Figure 1. Even in this worst case 
 note how closely the sample distribution function fits the hypothesized 
 distribution. 
 
 The K-S tests described above provide a check on independence in 
 the time domain; a check in the frequency domain was also performed. By 
 summing the Fast Fourier Transform of each of the shuffled sequences, 
 sample integrated periodograms were obtained. Under the independence 
 hypothesis, the integrated periodogram values should increase linearly 
 from zero to one as the frequencies increase from zero to one-half. The 
 Grenander-Rosenblatt (G-R) statistic may be used to measure the discrep- 
 ancy of the sample integrated periodogram from linearity. If the hypo- 
 thesis is true then a factor of V20U8 / \J2 = 32 times the maximum ab- 
 solute difference between the sample integrated periodogram and twice the 
 corresponding frequency should have the distribution calculated in Hannan 
 (1967) ; departures from independence will tend to make the sample sta- 
 tistics too large to fit the distribution. The five sample statistics 
 and corresponding P-levels are shown in Table 2 below along with the chi- 
 square summary statistic defined in the last section. Seed #1 had the 
 worst P-level so a graph of the corresponding sample integrated periodo- 
 £ram is included in Figure 2. 
 
 Computed with SOUPAC program FASPER available from Computing Services 
 Office, University of Illinois, Urbana, Illinois 618OI. 
 
 69 
 
8 
 
 en 
 
 o 
 
 o 
 
 CO 
 
 o 
 
 
 o 
 o 
 
 COM 
 
 ee*o 
 
 SL'Q 
 
 CETO 
 
 k.:o 
 
 &e*o 
 
 ;s'o 
 
 ti 
 
 o 
 
 04- 
 
 3 
 
 o 
 o t 
 
 o 
 O 
 
 I 
 
 70 
 
OOOOOOOOOOOOOCOOOOOOOCOOOOOOOOCOOOOOCOOCOOCOOOOGOOOOOOO 
 
 I I i , I . I I I I I 
 
 UJU-'U-A'JUiU L UJUJUiUHJ UJU LJUU-U U UjUili UU Uj... U u UJUIOi" OJU-LL UJUm U tUU.U'U-'liiUJUJUjUJUJULiUlUJUIU'UJ 
 
 ^r»rg^— '^oir>o>»C'(^a'(Mr-— O'-'irvo^r-f^ xiMf-rsjo^oo^rc**? •ri^r-rN.-o-^.oOinc'^ c<MCN^hNo^o 
 ^(^rg^NC M**C K ><<f>i | C , -<<»»)C'<*<;i-' r >'CX— i^^S-c.iroccr^^c or^incco<MiAr-o<\iir»r-.-<.coi/>.0 
 
 ©»0 , Occeoecacccaui^r"-f»^h»f»-^^^o.Cii"M/ v i<"vc">intf">»y ■f*?-t^-Tc<\r*r r n r \r~ fNf\i(\if\/rMr\ip— h« *— -•——>'J)r*\f\e>t->if\ 
 OOOCOOOOCOOOOOOOOCOOOOOOCcOOCOOOOOOOCOOOOCpOOOOOOOOOOQO 
 
 
 
 ON* 
 
 p* 
 />#• 
 
 • • 
 NO« 
 
 •H ♦ 
 
 i*. 
 
 ©* 
 o« 
 
 «\ • 
 
 K, £ 
 
 • • 
 
 fsiO# 
 
 «M * 
 
 «*\ * 
 
 ' E* 
 
 • • 
 
 I • • 
 * 
 
 ox 
 
 X 
 
 r-i* 
 
 r 
 
 3 
 
 § 
 
 ' 4 
 t 
 
 4 
 • 
 
 I • • 
 
 rte* 
 
 |0 8 
 
 £ 
 
 ► IT 
 
 s>* 
 
 
 r 
 < 
 a. 
 O 
 
 co 
 
 OK 
 
 *o 
 III 
 
 ►- 
 < 
 a" 
 O 
 
 UJ 
 
 2 
 
 
 I 
 
 co<v 
 
 r 
 r 
 
 00 f^ 
 
 IMfcC 
 
 z 
 r 
 
 xo 
 
 iAJ"k 
 
 x 
 
 X 
 
 # 
 
 I 
 
 ccooocoo?o^coocc;^o 
 coooroot'oc "CJC(Ooo^o 
 
 ! : I 
 
 li.lltllll! '-■II a-«"L.>i:-J.MiU>i'IU'l.:w>l. •; {. 
 m .*• j-oo— •!*■<•* t. o-oir — mwtj r-M. 
 l«l!Tj*-»^0""' np ^ »f"^~.»rv*"- — < J«-<'0 
 
 r-C'^ .»►-'» N>0'.' -s'.'.'-i.fj' — 
 
 •f M— < ~" s -'"y ^r- - ic *"t~-it. •'i — ir .j>. j- ~-». 
 
 CCi 
 oo- 
 
 x 
 x| 
 
 3T 
 X 
 
 mo 
 
 x 
 
 *»«■*(**** 
 
 C 
 
 x 
 
 (MOO 
 
 X 
 
 z 
 xo 
 n 
 
 <\J00 
 
 HttMU.M 
 
 £X 
 
 3 
 
 xai 
 u 
 
 * • » 
 
 COIN* 
 X 
 XX 
 
 c X: 
 
 * «■ » « 
 
 
 oc-o 
 
 CO 
 
 o 
 
 CO 
 C.J 
 
 T 
 
 y 
 
 
 
 ccaccjcioojocoaoocaooo 
 i i 
 
 i 
 
 
 Ull. I 
 
 
 
 ■4 -.' r . 
 
 • fill 
 
 c.i'i'iu u u.uin-ti u.ut'i u:'l 
 
 ^1 . • ^^ * ^ *^» - f\ '.»■ % *- T3 m ■ f . rxj — «r*^l T* 
 .■r --\ -^ .-^. *^ ^j <^j -g r> i pvj <%:-^— < ^i --* _4 ^^ ^ j **• *f\ **\ 
 
 
 II I 
 
 HIU 
 
 mo 
 
 OOjCOOO">00 JOQOOOOonO 
 
 i \ 
 
 ooqorocnooc'JOOQiooiooooiciooooooocoao 
 
 T 
 
 I 
 
 o 
 
 c 
 a> 
 
 & 
 
 w 
 
 <^i 
 
 (0 
 
 o 
 
 O 
 
 •H 
 
 <u 
 
 p. 
 
 -d 
 <u 
 
 is . 
 
 b0 
 
 a a> 
 
 CVJ 
 
 to 
 
 •H 
 
TABLE 2 
 
 
 Statistic 
 
 P-level 
 
 .1 
 
 2.21+3 
 
 .OU 
 
 2 
 
 .725 
 
 .88 
 
 3 
 
 1.578 
 
 .33 
 
 k 
 
 l.OOU 
 
 .63 
 
 5 
 
 .671 
 
 .92 
 
 -2*1 ln(p.) 
 
 1 
 
 10.00 
 
 .Ul» 
 
 The res\ilts of this series of tests, like those of the preceding 
 section, are quite satisfactory and give no cause to suspect interdependence 
 between simulation runs . 
 
 72 
 
D.3 INDEPENDENCE BETWEEN INPUTS TO A MONTE CARLO SAMPLE 
 
 While the frequency domain tests of the last section actually include 
 a test of the independence of sample inputs, further tests were performed 
 and are described below. The next two series of tests paralleled the two 
 series of the last section in methodology, but this time attention was 
 focused on the two sequences of 102U numbers generated from each of the five 
 seeds. In the time domain, the autocovariances at lags L = 1,512 of a sequence 
 were adjusted by a factor of /102U-L and tested with the Kolmogorov-Smirnov 
 statistic. Again, under the hypothesis of independence within each of the 
 ten sequences of 102^ numbers, the adjusted autocovariances of each sequence 
 should constitute a standard normal sample. The resulting sample statistics 
 and P-levels along with the chi-square summary statistic and its P-level are 
 shown in Table 3. Similarly, the ten sequences were tested for independence 
 
 in the frequency domain with the Fast Fourier Transform just as in the previous 
 section. The sample statistics and summary statistic are shown in Table k. 
 
 Again, the worst cases are illustrated for the time and frequency domain 
 
 tests in Figures 3 and k. 
 
 TABLE 3 
 
 
 Statistic 
 
 P-level 
 
 1A 
 
 IB 
 2A 
 2B 
 3A 
 3B 
 
 ua 
 
 kB 
 5A 
 5B 
 
 1.132 
 
 1.21U 
 
 1.077 
 
 .573 
 
 .882 
 
 1.106 
 
 .7^5 
 
 .657 
 
 .61*5 
 
 .U91 
 
 .15 
 .11 
 .20 
 .90 
 .U2 
 
 .17 
 .61* 
 .78 
 .80 
 .97 
 
 -2*1 ln(P.) 
 l 
 
 18.81 
 
 .53 
 
 73 
 
3 
 
 rj 
 
 CD 
 
 
 TU 
 
r 
 
 ,,.,, ...u ii, a. u iLu.uJii'UJU'U a u-U-iu UIU.U I'.U'U.IDU ujujUiu uiuu uju'UUiuju'u-uiu uiilu'liuiuui I 
 
 ■tU'00-(M(0^i.'>inONc:^c , CMNr v *ir<.'\of>-xroOHM<u^'/vON'rr-co-^i'^^(r.^irt',-0 i 
 
 ooooooocooooococcocooooooooooocooocooooococooooooooooo 
 
 • »m 
 
 ¥NO»X 
 
 ITMi 
 
 • • 
 
 «*•©# 
 m « 
 
 "* * 
 
 • f 
 
 o« i 
 
 O* I 
 
 **! 
 
 m «| 
 
 r» « 
 m * 
 
 • •' 
 ©m*| 
 
 a* I 
 
 (is! 
 
 • t, 
 mo* ; 
 m i 
 m * 
 m « 
 
 • 
 Or"* 
 
 o« 
 
 o# 
 m« 
 
 • • 
 (*>0-» 
 
 in * 
 
 m 
 
 A. 
 mm 
 
 m 
 
 X' 
 
 in 
 m i 
 
 in: 
 
 xm 
 m 
 
 m 
 
 nim 
 »n 
 mm. 
 *x 
 
 5- CM 
 
 m 
 
 m 
 
 o.-g», 
 
 o» 
 
 m# 
 
 m*; 
 
 • t 
 
 <NlO» 
 -n «. 
 m * 
 m * 
 • 
 
 o* 
 
 Oft 
 
 m* 
 
 • «, 
 mO* 
 m * 
 
 — * 
 
 • «t 
 
 o* 
 in* 
 
 HOI 
 
 tn p- 
 
 m s 
 
 -« * 
 
 • « 
 
 o« 
 
 OK 
 -** 
 
 • i 
 -Oi 
 
 •»in * 
 
 i»h- * 
 ♦ O * 
 *. • • 
 
 fOO« 
 
 e* 
 
 r uv 
 < o* 
 
 a • • 
 
 ooo* 
 cm * 
 cm « 
 
 CO * 
 
 ero 
 
 UJ 
 
 a 
 
 a. *. 
 
 o 
 
 o 
 
 UJ 
 
 m 
 
 m 
 
 mf>-, 
 m«\i 
 tn 
 
 x«*- 
 ' o 
 
 «*-x 
 
 in 
 
 *x 
 n 
 
 n 
 
 ■tf-m 
 in 
 in 
 
 mm 
 
 x>* 
 
 o 
 
 X.J- 
 
 mm 
 m 
 
 LI 
 
 mm 
 in 
 
 o 
 
 mm 1 
 >n 
 
 *x 
 
 m 
 
 x>t 
 .n 
 
 in 
 in 
 
 tn 
 
 X«f 
 
 in 
 x*r 
 
 m 
 
 mm> 
 
 xy 
 
 ;mm 
 mm 
 
 >rx 
 
 o 
 
 Li 
 X. 
 
 ►X | 
 
 w-l I 
 
 mr^ 
 
 h*x 
 .n 
 
 «j-x 
 m 
 
 « 
 * 
 « 
 
 « 
 
 « 
 
 •J 
 
 « 
 
 * 
 
 *rx 
 
 mm 
 
 
 booc'cncrr ocr>r »?">c c itiror it c-r-ocoorjc oooooooo oo^oo 
 
 ;ccoo?oo?c;ccoooccooocjc 
 
 I 
 
 'a-H'ji mm u. l U' J i.l. j UiuiI:<l.u.j i' ' ul u_i_ l-i. u-uit'U'UiU u uji:.u u,uri_..uu i-.-iu 
 
 ffif 1 ci>-i'« O'.'-'ncw »>»i ?!•»'». 'cc- •mm^"».'«L* j'- i i^v. ; v — <m-^ r-.*>i">^.- — ;?-:—— « 
 Jm o r ~j <v- "n. .-* -*»«.rr~ -r? i-r ■■" ^j-~- "*r ,.* r — . -f- 7 ^^ "»m.r -v- »." > j»—-r^-' — •<. r - •" .- " 
 
 (M* V i- — n;f^f^f^r^(-~r^ oo^j • o >. ■>>•" >-" x -* Oi "W vT >T >r r -Tfi iWitiHi^NNNMM'JH'' 
 
 "(JOOOC OO DO 
 I I I I I I 
 
 a. i' UIUIL.1..1.H1 j.uj 
 
 00000000 r 0000000 00 3o,oocjoo-j 00 J ooooooocooqo^o 00 c -3100000. ^00 
 
 <\J 
 
 3! 
 
 I 
 
 o 
 
 g 
 
 a; 
 
 CO 
 
 -p 
 to 
 
 >H 
 •H 
 
 o 
 
 Pi 
 
 o 
 
 u 
 o 
 
 •H 
 
 CD 
 Pi 
 
 cu 
 
 13 
 
 w 
 
 CD 
 
 •H 
 
TABLE h 
 
 ' 
 
 
 
 
 Statistic 
 
 P-level 
 
 1A 
 
 .659 
 
 .93 
 
 IB 
 
 2.259 
 
 .05 
 
 2A 
 
 2.1+39 
 
 .03 
 
 2B 
 
 l.U6l 
 
 .29 
 
 3A 
 
 .602 
 
 .96 
 
 3B 
 
 1.52U 
 
 .25 
 
 UA 
 
 1.U23 
 
 .31 
 
 ub 
 
 .910 
 
 .71 
 
 5A 
 
 .866 
 
 .75 
 
 5B 
 
 .833 
 
 .79 
 
 -2*Z in (P.) 
 
 22.55 
 
 .31 
 
 These results, like those of the preceding section are quite 
 acceptable. Overall then, the independence properties of GGNRF are 
 very satisfactory. 
 
 76 
 
D.U TESTING FOR NORMALITY 
 
 A final series of tests was undertaken to examine the suitability 
 of GGNRF in an application calling for normally distributed random 
 numbers. First, the Kolmogorov-Smirnov goodness of fit test was applied 
 to the ten sequences of 102U numbers with the results shown in Table 5 
 Following the same format as in previous sections, the sample statistics 
 and their P-levels are given along with the summary chi-square statistic, 
 The plot of the sample distribution function with the worst fit is shown 
 in Figure 5. 
 
 TABLE 5 
 
 
 Statistic 
 
 P-level 
 
 1A 
 
 1.210 
 
 .10 
 
 IB 
 
 .799 
 
 .55 
 
 2A 
 
 .70U 
 
 .70 
 
 2B 
 
 .578 
 
 .89 
 
 3A 
 
 1.111 
 
 .17 
 
 3B 
 
 1.187 
 
 .12 
 
 kA 
 
 1.095 
 
 .18 
 
 UB 
 
 .590 
 
 .88 
 
 5A 
 
 1.053 
 
 .22 
 
 5B 
 
 .609 
 
 .85 
 
 -2*1 ln(P.) 
 l 
 
 21.57 
 
 .36 
 
 77 
 
o 
 o 
 
 GO - 1 
 
 78 
 
Although these results indicate good fit , especially given the 
 relatively large sample size, the sample variances and means were also 
 checked. Given the standard normality hypothesis, the sample variances 
 multiplied by 1021+ should be chi-square distributed with 1023 degrees 
 of freedom. The sample variances and the corresponding P-levels under this 
 hypothesis are listed below in Table 6 along with the usual summary 
 statistic. . 
 
 TABLE 6 
 
 
 Statistic 
 
 P-level 
 
 1A 
 
 .979 
 
 .77 
 
 IB 
 
 .915 
 
 .97^ 
 
 2A 
 
 1.106 
 
 .009 
 
 2B 
 
 .936 
 
 .925 
 
 3A 
 
 .997 
 
 .51 
 
 3B 
 
 1.038 
 
 .19 
 
 HA 
 
 1.033 
 
 .22 
 
 Ub 
 
 1.022 
 
 .30 
 
 5A 
 
 .963 
 
 .79 
 
 5B 
 
 .998 
 
 .50 
 
 -2*1 ln(P.) 
 
 22.11 
 
 .33 
 
 79 
 
The sample means should be normal with mean and variance 1/1021+ . 
 Multiplying the sample means by a factor of 32 should result in numbers 
 drawn from a standard normal population. However, the large size of 
 the P-levels in Table 7 gives cause for suspicion that the hypothesis 
 is not true. On the other hand, the sample size of 1021+ was originally 
 chosen to be large enough to signal even acceptably small deviations from 
 ideal behavior; the very worst sample mean was only -.062. Furthermore, 
 given the amount of testing undertaken in this study, it is to be expected 
 that sooner or later some test results will go awry. To shed more light 
 on the matter, further tests of the mean tendency of GGNRF seemed appro- 
 priate. Since only one seed would be needed for the actual Monte Carlo 
 application, seed #3 was selected for a more intensive examination. 
 Eleven thousand numbers were drawn from seed #3 and then discarded in 
 order to skip over the two strings of 1021+ numbers previously tested. 
 Then ten consecutive strings of 102U numbers were drawn and their sample 
 means were computed. The results are listed in Table 8. In addition, 
 ten new seeds were selected and samples of 1021+ numbers drawn from each. 
 The data for these sequences is shown in Table 9. 
 
 TABLE T 
 
 
 Statistic 
 
 P-level 
 
 1A 
 
 -.01+9 
 
 .9* 
 
 IB 
 
 -.019 
 
 .72 
 
 2A 
 
 .0083 
 
 .1+0 
 
 2B 
 
 .0069 
 
 .1+1 
 
 3A 
 
 .031+ 
 
 .11+ 
 
 3B 
 
 -.01+2 
 
 .91 
 
 UA 
 
 -.067 
 
 .98 
 
 UB 
 
 -.021 
 
 .75 
 
 5A 
 
 -.056 
 
 .95 
 
 5B 
 
 -.019 
 
 .72 
 
 -2*1 1 (P ) 
 
 9.89 
 
 .97 
 
 n 1 
 
 
 
 An 
 
TABLE 8 
 
 
 1 
 
 Statistic 
 
 P-level 
 
 
 1 
 
 2 
 3 
 
 b 
 
 5 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 .035 
 .013 
 -.029 
 -.0035 
 -.036 
 -.016 
 -.005^ 
 .025 
 .06b 
 -.0060 
 
 .13 
 .3U 
 .82 
 
 .5h 
 .87 
 • 70 
 .57 
 .21 
 .02 
 .58 
 
 
 -2*1 ln(P.) 
 
 21.925 
 
 ,3k 
 
 TABLE 9 
 
 
 Statistic 
 
 P-level 
 
 1 
 
 .OUl 
 
 .10 
 
 2 
 
 .0053 
 
 .U3 
 
 3 
 
 -.0U 
 
 .90 
 
 1* 
 
 .023 
 
 .23 
 
 5 
 
 -.010 
 
 .63 
 
 6 
 
 -.02U 
 
 .78 
 
 7 
 
 .029 
 
 .17 
 
 8 
 
 .018 
 
 .28 
 
 9 
 
 .0026 
 
 M 
 
 10 
 
 -.0035 
 
 .5b 
 
 -2*Z ln(P.) 
 
 19.1b 
 
 .U8 
 
 81 
 
The results of these tests certainly do much to allay the suspicions 
 raised by the results in Table 7. Neither would it seem that seed 03 
 has run into an area of systematically bad behavior (witness Table 8), nor 
 would it seem that there is an overall bias in the generator (witness Table 
 9). These tests together with the preceding K-S tests and sample variance 
 tests indicate that GGNRF has satisfactory distributional properties. 
 Overall then, GGNRF tests out as a satisfactory normal random number 
 generator. 
 
 82 
 
REFERENCES 
 
 1. C. Bullard and A. Sebald, "Effects of Parametric Uncertainty and Technological 
 Change in Input-Output Models," CAC Document No. 156, Center for Advanced 
 Computation, University of Illinois, Urbana, IL 6l801, March 1975. 
 
 2. C. Bullard, A. Sebald, D. Putnam, D. Amado , "Stochastic Analysis of Un- 
 certainty in a U.S. Input-Output Model," CAC Document (forthcoming). 
 
 3. C. Bullard and R. Herendeen, "Energy Cost of Consumption Decisions", 
 Proc. IEEE, March 1975- Also available as Document 135, Center for 
 Advanced Computation, University of Illinois , Urbana 
 
 IL 6l801. 
 
 h, C. Bullard, "Uncertainty in the 1967 U.S. Input-Output Data," CAC Document 
 No. 191, Center for Advanced Computation, University of Illinois, 
 Urbana, IL 618OI, April 1976. 
 
 5. H. Cramer, Mathematical Methods of Statistics , Princeton University 
 Press, Princeton , NJ, 19*+8. 
 
 6. Hannan, E. J., Time Series Analysis , Science Paperbacks and Methuen & Co. 
 LTD, London, 1967. 
 
 7. W. Hayes and R. Winkler, Statistics: Probability, Inference and Decision , 
 Holt Rinehart and Winston Inc., New York, 1970. 
 
 8. B. Jansson, Random Number Generators , Victor Pettersons Bokindustri 
 Aktiebolag, Stockholm, 1966. 
 
 9. R. Knecht, "Reliability Measures of CAC-ERG Direct Energy Use Data," 
 
 CAC Technical Memorandum No. 67, Center for Advanced Computation, University 
 of Illinois, Urbana, IL 6l801, November 1975- 
 
 10. H. Kuki, "A Fast Normal Random Number Generator", Computation Center, 
 The University of Chicago, Chicago, IL, 197*+. 
 
 11. W. Leontief , The Structure of the American Economy 1919-1939 , Oxford Univer- 
 sity Press, New York, 19 Ul. 
 
 12. 0. Morganstern, On the Accuracy of Economic Observations , Princeton University 
 Press, Princeton, NJ, 1950. 
 
 13. A. Sebald, "An Analysis of the Sensitivity of Large Scale Input-Output 
 Models to Parametric Uncertainties," CAC Document No. 122, Center for 
 Advanced Computation, University of Illinois, Urbana, IL 6l801, November 197*+. 
 
 lU. D. Smith & D. Simpson,' "Direct Energy Use in U.S. Economy, 1967," CAC 
 
 Technical Memorandum No. 39, Center for Advanced Computation, University of 
 Illinois, Urbana, IL 6l801, January 1975. 
 
 83 
 
15. M. A. Stephens, "Asymptotic Results for Goodness-of-Fit Statistics with 
 Unknown Parameters", Annals of Statistics , V.k, no. 2, 1976, pp. 357-369. 
 
 16. U.S. Department of Commerce Bureau of Economic Analysis, Input-Output 
 Structure of the U.S. Economy 1967 > U.S. Government Printing Office, 
 Washington, D.C., 197^. 
 
 17. U.S. Department of Commerce Bureau of Economic Analysis, Definitions and 
 Conventions of the 1967 Input-Output Study , (mimeo) 197*+. 
 
 18. S. S. Wilks , Mathematical Statistics, John Wiley & Sons, New York, 1962. 
 
 8k